## Partial Derivative Questions and Answers

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Use the definition of partial derivatives as limits to find f_x(x, y) and f_y(x, y). f(x, y) = xy^2 - x^3y

The cost of renting a car from a certain company is $42 per day plus 24 cents per mile, and so we have C = 42d + 0.24m. Find \dfrac{\partial C}{\partial d} and \dfrac{\partial C}{\partial m}. Give...

Find the partial derivatives f_xxx(x,y) and f_xyx(x,y) for f(x,y) = (x^9)(y^9) - (x^8)(y^8)

Find f_x, f_y, f_{x,y}, f_{xx},f_{yy} where f(x,y) = x^2 + 5xy + \sin x + 7e^x

Use the Chain Rule to find \frac{\partial z}{\partial s} and \frac{\partial z}{\partial t}. z = \ln\left(2x^2 + 3y\right), \enspace x = t\sin s, \enspace y = t^2\cos s

Consider the following function: f(x, y) = x^y. Find f_y.

Consider the following function: f(x, y) = x^y. Find f_x.

Evaluate the following integral using Partial Fraction Decomposition: \int \dfrac{8}{(x - 2)(x + 6)} \: dx.

Consider the function f(x, y) = e^{xy}. Calculate f_{xyy}.

If f(x, y, z) = -xzy + (e^(3y))/((zy) + 5y^2 + 5), then (partial^3 f)/(partial z partial y partial x) at the origin = A. -1 B. 0 C. 1/5 D. None is true.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

a. Find the partial of f with respect to y and the partial of f with respect to x for f(x,y) = x^2 + y^2. b. Find the partial of f with respect to y and the partial of f with respect to x for f(x...

Compute the partial derivative f_y(0, \pi) for f(x, y) = \sin\left(x^3 - 6y\right).

Given F(r, s, t) = -r\left(7s^6 + 2t^5\right), compute F_{rst}.

Let f(x, y) = arctan(y/x). Evaluate partial f/partial y at (x, y) = (5, 7). Your answer will take the form a/b where a is an integer, b is a positive integer, and a and b have a common factor of 1.

Determine the partial derivatives f_x and f_y of the function f(x,y) = x^y

Which of the following is the gradient vector field of f(x, y) = e^x \cos 2y + 2x - y^2? (a) \nabla f(x, y) = \bigl \langle e^x \cos 2y + 2, -2e^x \sin 2x - 2y \bigr \rangle (b) \nabla f(x, y) = \b...

Given that z = x^2 + y^3, x = 2u^2 - v^2, and y = sqrt(2u - 1) + v^2, find the partial of z with respect to v at (u,v) = (1,-1): a. -20 b. None c. 20 d. -16 e. -96

Find the gradient vector field of f(x, y, z) = x + yz + cos y.

\dfrac{\partial}{\partial v} \left(\tan^{-1}(vx)\right) =

If f(x, y) = 7xy + e^{7xy}, find F_{xx} and F_{yy}.

Calculate all four second-order partial derivatives of f(x, y) = (3x + 5y) e^y.

For the function z(x, y) = \tan^{-1}\left(x^2y\right) + \tan^{-1}\left(xy^2\right), find z_x and z_y.

Let f(x, y, z) = x^2 + 2xy - z^3. Then the gradient nabla f at (1, 1, 1) is: a. 4i + 2j - 3k b. 4i - j c. 4i + j d. None

For z = e^{xy} with x = 2t + s and y = \dfrac{t}{s}, find \frac{\partial z}{\partial t} \Big|_{t = 1, \, s = -2}. Enter your answer as an integer (no decimal point).

Find the second partial derivative (partial^2f(x,y) over partial x^2) of the function f(x,y) = 3xy e^-x^2 - y^2.

Given f(x, y) = y^3e^{-5x}. Find \frac{\partial^3 f }{\partial y \partial y \partial x} at (0,1). Select one: a. -30 b. -125 c. 50 d. 150

Find \frac{\partial f}{\partial x} and \frac{\partial f}{\partial y} for f(x, y) = (xy - 1)^2.

Consider z = r^3\theta + e^{r + \theta^2} + \theta \sin r. \dfrac{\partial z}{\partial \theta} = \\ \dfrac{\partial^2z}{\partial r \partial \theta} =

Find \frac{\partial f}{\partial x} and \frac{\partial f}{\partial y} for f(x, y) = \dfrac{x}{x^2 + y^2}.

If h(s, t) = \dfrac{s^2 + 4}{t - 3}, then h_t(s, t) = (a) \dfrac{2s}{t - 3} (b) \dfrac{s^2 + 4}{(t - 3)^2} (c) \dfrac{2s(t - 1) - \left(s^2 + 4\right)}{(t - 3)^2} (d) s^2 + 4 (e) -\dfrac{s^2 + 4}{...

Find the gradient of the scalar field V(x, y, z) = xy + xz^3y.

Compute the partial derivative of f(x, y) = x^8 + e^y with respect to r where: x(r, s) = s cos (5r) y(r, s) = 3s + r

Prove that y = In(d(x-vt)) is a solution of the standard wave equation, where d is a constant.

Given that z = x^2 + y^3, \enspace x = 2u^2 - v^2, \enspace y = \sqrt{2u - 1} + v^2. Find \frac{\partial z}{\partial v} at (u, v) = (1, -1) (a) None (b) 20 (c) -20 (d) -96

Find the maximum rate of change of f(x, y, z) = x + \dfrac{y}{z} at the point (-2, 2, 2) and the direction in which it occurs.

Let f(x , y) = e^{x^2 y} \tan(x + y). Find \frac{\partial^2 f}{\partial y \, \partial x}.

A monopolist manufactures and sells two competing products, call them I and II, that cost $45 and $37 per unit, respectively, to produce. The revenue from marketing x units of product I and y units...

True or false? Give reasons for your answer. There is a function f(x, y) with f_x(x, y) = y^2 and f_y(x, y) = x^2.

Consider the Cobb-Douglas Production function: P(L, K) = 11L^{0.3}K^{0.7}. Find the marginal productivity of labor (that is, P_L) and marginal productivity of capital (that is, P_K) when 12 units...

Let \phi(x, y, z) = x^2y^2 + y^3z^3 + zx. Find \text{div}(\text{grad}\, \phi). a. 0 b. 2y^2 + z + 6yz^3 + 2x^2 + x c. 2y\left(y + 3z^3\right) d. 2y^2 + 2x^2 + 6yz^3 + 6y^3z e. \left(2y^2, 2x^2...

Find all second-order partial derivatives of z = f(x, y) = e^{x^2 + y^2} + \frac{x}{y}.

Consider the Cobb-Douglas Production function: P(L, K) = 30L^{0.2}K^{0.8}. Find the marginal productivity of labor and marginal productivity of capital functions.

If f(x, y) = e^xy + x^2y + \cos y, find (a) f_x (b) f_y (c) f_{xy}

Given f(x,y,z) = sqrt (x^2 + y^2 + z^2), find the maximum rate of change of f at the point (3,6,-2) and the direction in which it occurs.

Find the maximum rate of change of at the point (2, 1). In which direction does it occur?

Let f(x, y) = \dfrac{\ln\left(y^2 + 1\right)}{x}. What is f_y(x, y)? (a) \dfrac{\ln\left(y^2 + 1\right)}{x} (b) -\dfrac{\ln\left(y^2 + 1\right)}{x^2} (c) \dfrac{1}{x\left(y^2 + 1\right)} (d) \dfrac...

Let f(x, y) = \dfrac{x}{y^2} - \dfrac{y}{x}. What is f_x(x, y)? (a) \dfrac{1}{y^2} + \dfrac{y}{x} (b) \dfrac{1}{y^2} + \dfrac{y}{x^2} (c) \dfrac{x^2}{2y^2} + \ln y (d) -\dfrac{2x}{y^3} - \dfrac{1}{...

Calculate \dfrac{\partial^2 u}{\partial t^2} for u = \cos(10x - 5t).

A monopolist manufactures and sells two competing products, call them I and II, that cost $31 and $17 per unit, respectively, to produce. The revenue from marketing x units of product I and y units...

Find (\partial f/ \partial x)_y and (\partial f/ \partial y)_x for the following function, where a,\, b are constants. f = a\cos^2(bxy)

Find (\partial f/ \partial x)_y and (\partial f/ \partial y)_x for the following function, where a,\, b are constants. f = a\exp\left(-b\left(x^2 + y^2\right)\right)

f(x, y) = x^2 + 4xy + 2y^4 Find the exact points (x, y) where f(x, y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature of f(x, y...

f(x, y) = 2x^2 - 2y^3 - 12x + 24y + 18 Find the points (x, y) where f(x, y) has a possible relative maximum or minimum. (Give the exact points.)

Find the first partial derivatives of \mathbf{v}_1 = e^x \cos y, e^x \sin y and \mathbf{v}_2 = \cos x^2y, -\sin xy^2.

f(x, y) = 3x^2 + y^2 - 12x - 8y + 28. Find the point (x, y) where f(x, y) has a possible relative maximum or minimum. (Give the exact point.)

f(x, y) = x^2 - 2xy + 3y^2 + 4x - 32y + 57. Find the exact point (x, y) where f(x, y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the...

f(x, y) = x^2 + 4xy + 2y^4. Find the exact points (x, y) where f(x, y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature of f(x,...

Find the indicated partial derivatives. f(x, y) = x^2y - 4x + 3\sin y: \enspace \frac{\partial^2f}{\partial x^2},\,\frac{\partial^2f}{\partial y^2},\,\frac{\partial^2f}{\partial y \partial x}

True or false? A partial differential equation of order 2 is linear.

Find all first-order partial derivatives. f(x, y) = x^2\sin xy - 3y^3

Prove that the function u = e^{-2x}\sin 2y - e^{-3y}\cos 3x satisfies Laplace's equation u_{xx} + u_{yy} = 0.

Prove that the function z = f(x, y) = 1.2x^{0.6}y^{0.4} satisfies the equation x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = z.

Find \frac{\partial z}{\partial x} if z = 7x^3 + 3x^2y + 2y^3.

Find the partial derivatives \frac{\partial w}{\partial r}, \, \frac{\partial w}{\partial s}, \, \frac{\partial w}{\partial t} of the function w = 3r^2 + 1s^2 + 5t^2.

The volume V (measured in cubic metres) of a certain amount of gas is determined by the temperature T (measured in Kelvin) and the pressure P (measured in megapascals) by the formula V = 0.08(t/p)....

The volume V (measured in cubic meters) of a certain amount of gas is determined by the temperature T (measured in Kelvin) and the pressure P (measured in megapascals) by the formula v = 0.08(T/P)....

f(x,y,z) = \frac{x}{2y} + 5z What are the partial derivatives for x,y, and z for (2,3,4)?

From the equation z \cos (z) = x^3y^3 + z, where z is a function of x,y, find \frac{\partial z}{\partial x}

Let f(x, y, z) = x^{2} - 2 y^{2} / y^{2} + 3 z^{2}, find the following a) f_x (x, y, z) b) f_y (x, y, z) c) f_z (x, y, z)

If f(x, y) = x^3 + 5y^2, find the partial derivative of f with respect to y.

If f(x, y) = x + 2y^2, find the partial derivative of f with respect to x.

Find f_{yz}(x, y, z) and f_{xz}(x, y, z) if f(x, y, z) = \sin(2yz + 6xz). Find f_{yz}\left(0, \frac{\pi}{4}, 1\right) and f_{xz}\left(\frac{\pi}{3}, 0, 1\right).

Find the partial derivatives partial f/partial y, partial f/partial z, if they exist, of the function f(x, y, z) = 3x/(1 + cos^2(x^3 y)) e^(x^2 z).

The maximum rate of change of the function f(x, y) = \ln\left(x^2 + y^2\right) at the point (-1, 1) is: (a) 2 (b) 1 (c) -1 (d) \sqrt{2} (e) -\sqrt{2} (f) -2

Let z be implicitly defined as a function of x and y through the equation y^3z + x\ln y = z^5. Find \frac{\partial z}{\partial x} and \frac{\partial z}{\partial y}.

Let f(x, y, z) = 8x^4y + 2xy^5 + 3yz^7. Find the iterated partials f_{xy},\, f_{yz},\, f_{zx},\, f_{xyz}. (Use symbolic notation and fractions where needed.)

Let f(x, y) = 6x^2y^3. Then f_x(x, y) = \rule{2cm}{0.4pt} \\ f_x(3, y) = \rule{2cm}{0.4pt} \\ f_x(x, -5) = \rule{2cm}{0.4pt} \\ f_x(3, -5) = \rule{2cm}{0.4pt} \\ f_y(x, y) = \rule{2cm}{0.4pt} \\ f_...

Let f(x, y) = 2xy^3 + 3x^2y^2. Then f_{xy} = (a) 2y^3 + 6xy (b) 6y^2 + 12xy (c) 6y^2 (d) 12xy + 6x^2

Given f(x, y) = e^{x^2 - 4y^3}, find \frac{\partial^2 f}{\partial x \partial y}.

Compute the first-order partial derivatives \dfrac{\partial w}{\partial x}, \dfrac{\partial w}{\partial y}, and \dfrac{\partial w}{\partial z} of the function w = \dfrac{7x}{\left(x^2 + y^2 + z^2\r...

Find the indicated partial derivatives using the Chain Rule.

Compute the first-order partial derivatives of the function. z = e^{-x^8 - y^5} (Use symbolic notation and fractions where needed.) \dfrac{\partial z}{\partial x} = \\ \dfrac{\partial z}{\partia...

Find the directions in which the function f(x, y) = x^2 + xy + y^2 increases and decreases most rapidly at P_0 (-4, -2). Then find the derivatives of the function in these directions.

Find the local maximum and minimum values and saddle point(s) of the function. f(x, y) = y^3 + 3x^2y - 6x^2 + 2

Find all the local maxima, local minima, and saddle points of the function. f(x,y) = x^2 + xy + y^2 + 4x - 4y + 2

A function u(x,y) satisfies Laplace's Equation if u_{xx}+u_{yy} = 0. Verify that the function u(x,y) = x^2-y^2+e+1 satisfies Laplace's Equation.

Find the indicated partial derivative. Please enter exponents as fractions and not as decimal numbers; for example, write x^{\frac{3}{2}} rather than x^{1.5}: \dfrac{\partial^2}{\partial x \partial...

Find the indicated partial derivative: \dfrac{\partial^2}{\partial y^2} \left(3x^3y^2 - 2xy^3\right)

Find the first partial derivatives of the function with respect to x and y. f(x, y) = e^{xy + 6} f_x(x, y) = __________ f_y(x, y) = __________

Find the first partial derivatives of the function with respect to x and y. f(x, y) = \dfrac{2x}{y^2} \\ f_x(x, y) = \\ f_y(x, y) =

Find the second partial derivative \left(\frac{\partial^2y}{\partial x^2}\right) of the function f(x, y) = 3x - x^4 - 4y^2 - 10xy.

The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Find the partial derivatives. del P/del V, del V/del T,...

Find all the second-order partial derivatives of the following function. w = 2x^2 tan(8x^5 y).

Find all the second-order partial derivatives of the following function. w = 2x^2 \tan (2x^3y)

Find all the second-order partial derivatives of the following function. w = 8x^2 \tan(7x^2y)

Using the following equation, prove the conclusion of Clairaut's Theorem.

Which of these functions (or none of them): (1) z = e^y \left ( y \cos x - x \sin y \right ) or (2) z = e^y \left ( y \cos x - x \sin x \right ) satisfies the equation \frac{\partial^2 z}{\parti...

Find the value using given equation.

Using the given equation, find the value. Hint: What is the easiest order of differentiation?

From the equation z \cos (z) = x^3y^3 + z, where z is a function of x,y, find \frac{\partial z}{\partial y}

Determine all first and second partial derivatives of the following function. g(x,y,z)=e^{xz}+3x^2sin(xy)+cos(z)

Verify the conclusion of Clairaut's Theorem.

Using the following, give the indicated partial derivative or derivatives.

Give the partial derivative or derivatives of the following.

Find the indicated partial derivative or derivatives.

Give the indicated partial derivative or deritatives.

Give the indicated partial derivative(s).

Given that \phi(x, y, z) = xyz - xy^2z^3, find grad \phi at the point P(-1, 1, 1). a. (1, 1, 2) b. (0, 1, 2) c. (0, -1, 2) d. (1, 0, 2) e. (1, -1, 2)

If f(x, y) = \sqrt3{x^3 + y^3}, find f_x(0, 0).

Find \frac{\partial^2 z}{\partial x \partial y} for z = \frac{\sin(2x + 3y)}{\sqrt{x^2 + y^2}}.

Find partial z over partial x and partial z over partial y by implicit differentiation if ye^x - 5 sin 3z = 3z.

Given z(x, y) = 5 cos(x^5 y^3), find partial z/partial x. Give your answer as a function of x and y.

If f(x, y) = tan^(-1)(y/x). Show that f_xx + f_yy = 0.

If f(x,y,z) = e^{x y z} then find f_{z z} at (1,-1,-1). a) e b) -e c) None d) 0

If we have the function f(x,y)= 1+ ln (3-x^2-y^2), then find partial f/partial t whenever x(t) (t+2)^2 and y(t)= (2/t^3).

At an Oregon fiber-manufacturing facility, an analyst estimates that the weekly number of pounds of acetate fibers that can be produced is given by the function: z=f(x,y) =1300x+3500y+7x^2y-13x^3...

Find all the second-order partial derivatives of g(x,y) = x^5y + 2 \sin (y) + 2y \cos (x).

Classify the following differential equation as either, elliptic, hyperbolic or parabolic. e^2x partial^2 u/partial x^2 + 2e^{x+y} partial^2 u/partial x partial y + e^{2y} partial^2 u/partial y^2=0.

Classify the following differential equation as either, elliptic, hyperbolic or parabolic. 3 partial^2 u/partial x^2 + 2 partial^2 u/partial x partial y + 5 partial^2 u/partial y^2+ 2 partial u/pa...

Find \frac{\partial z}{\partial x} using implicit differentiation. yz = \ln (x + z)

The gradient of f(x,y,z) = y \ln (x + y + z) at the point (-3, 4, 0) is _____

Given z = \sin (3x + 2y), show that 3 \frac{\partial^2 z}{\partial y^2} - 2 \frac{\partial^2 z}{\partial x^2}= 6z using partial derivatives.

Given z = 3x - x^2y^2, solve for \frac{\partial z}{\partial x} and \frac{\partial z}{\partial y} using partial derivatives.

Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 2 - 2x + 2y - 4x^2y at point (3,1). (a) Find f_x at (3,1) (b) Find f_y at (3,1)

Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 2 - 6x + 6y - 2x^2y at a point (2,3) (a) Find f_1 (2,3) (b) Find f_2 (2,3)

Find all the second-order partial derivatives of g(x,y) = x^5y + 4 \sin (y) + 2y \cos (x). \frac{\partial^2 g}{\partial x^2} = \frac{\partial^2 g}{\partial y \partial x} = \frac{\partial^2 g}{\pa...

Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 4 - 3x + 5y - 4x^2y at a point (3,3). (a) Find f_1(3,3) (b) Find f_2(3,3)

Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 7 - 5x + 2y - 2x^2y at a point (3,4). (a) Find f_1(3,4) (b) Find f_2(3,4)

Find S_{xy}(-1,1), if S(x,y) =x^3 ln y+4y^2e^x.

Let f(x,y)= e^y \sin (xy) Find f_{yx}

Find f_{yy}(x,y), if f(x,y) =(1+2xy^2)^8.

Given that f(x, y) = 5 x^{2} - 5 x^{2} y^{3} - 3 y^{4}, find f_{x} (x, y), f_{y} (x, y), f_{x x} (x, y), f_{x y} (x, y).

If f(x, y, z) = e^{x y z}, then find f_{x z} at (1,-1,-1). a. -e b. 0 c. e d. None

Let z = sin 2 x cos 3 y, x = t + s, y = s - t, then find z_{s} at (t, s) = (0,0). a. 1 b. 2 c. None d. -1 e. 0

Find the first partial derivatives of the function. f(x,y) = 4e^xy + 6

Let y = ln cube root of (x^2 + 2z^3 + cos(b^3)), find partial y/partial b.

Determine the partial derivatives of f_x and f_y of the function: f(x, y) = x^y.

Find the maximum rate of change of the function f(x, y) = 3x^3 - 2y^2 + xy at the point (2, 0).

Find f_{x y} (x, y ) using multivariate calculus. Given that f(x, y) = x^{4} y^{3} + 20 x^{2} y^{2}.

Find the first partial derivatives of the function. z = xy/(x^2 + y^2)

Find the first partial derivatives of the function. z = x^3 - 4xy + y^3

Find the first partial derivatives of f(x, y, z) = z arctan (-3y/2x).

The kinetic energy of a body with mass m and velocity v is . Show that

If z = ln (sqrt(x^2 +y^2)) find x (del z/del x) + y (del z/ del y)

Suppose z = 2{{x}^{2}}y-3xy+5 sin xy. Evaluate del^2 z/del x del y.

Find del^2 z/((del x) (del y)) when z = e^(x^3 +y^2).

Find del^2 z/del^2 x when z = e^(x^3 +y^2) .

Find the total differential of f(x, y) = ye^(2x) + sin(14x).

Find the gradient of the function f(x, y) = 3xe^(xy^3) + 6cos(y^3).

Use the Chain Rule to find del w/del t by if w = xyz + ln(xyz), x = t/s, y = s^3+1/t^2 , z = 8+ 1/t.

Given f(x,y) = e^(x^2 - 4y^3). Find (del f)^2/(del x)(del y).

Assume that all the given functions are differentiable. If z = f(x - y), show that (del z/del x) + (del z/del y) = 0

Find del z/del x and del z/del y. x^2 + 2y^2 + 3z^2 = 1

Find del z/ del x and del z/del y. e^z = xyz

If f(x, y) = x^2 sin(y) + y^2 cos(x) then partial f/partial y is _____.

If f(x, y) = (x - y)/(x + y) then partial f/partial x is _____.

Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2y, x = s + 2t - u, y = stu^2; del z/del s, del z/ del u, del z/del t when s=4,t=2,u=1

Use the Chain Rule to find del z/del s and del z/del t. z = x^2y^3, x = s cos t, y = s sin t

Find all the second partial derivatives. w = sqrt(u^2 + v^2)

Find all the second partial derivatives. v = e^((x)(e^y))

Find the indicated partial derivative. f(x,y,z) = y/(x + y + z);fy(2,1,-1)

Find the indicated partial derivative. f(x, y) = arctan(y/x); fx(2, 3)

Find the first partial derivatives of the function. u = sqrt(x1^2 + x2^2 + ......+Xn^2)

Find the indicated partial derivative. f(x,y,z) = sqrt(sin^2 x + sin^2 y + sin^2 z);f(z)(0,0,pi/4)

Find the first partial derivatives of the function. u = xy sin-1(yz)

Find the first partial derivatives of the function. w = ze^xyz

Find the first partial derivatives of the function. f(x, y, z) = x sin(y - z)

Find the first partial derivatives of the function. F(alpha, beta) = integral of sqrt(t^3 + 1)dt from alpha to beta

Find the first partial derivatives of the function. R(p, q) = tan-1(pq^2)

Find the first partial derivatives. w(u,v) = e^u over u^2 + v

Find the first partial derivative. w(u, v) = sin (u cos v)

Find the partial derivatives f_x and f_y of the function f(x, y). The variables are restricted to a domain on which the function is defined. f(x, y) = 6x^2 + 6xy + 5y^3.

If y = 3/42 x^7 - 10x^3 + 13x^2 + 28x - 2.7463 find d^3 y/dx^3.

If f(x, y) = ((x - 1)^3 + y)^(1/3), then f_x(1, 0) = _____.

Find the total differential. w = 7x + y over 2z - 3y

Find f_x, f_y, f_z, and f_yz for the following. Also find f_y(3, 4, -3) and f_yz(2, -2, 0). f(x, y, z) = 5x^5 + yz + 4z^4.

Find the partial derivative. Find f_y (6, 9) when f(x,y) = ye^{xy}. Leave your answer in terms of e.

Find nabla f at the given point. f(x, y, z) = x^2 + y^2 - 3z^2 + z ln x, (1, 1, 5).

Find partial derivative of f with respect to x and partial derivative of f with respect to y for the following function. f(x, y) = e^{8xy} ln (3y)

f(x,y) = (y^2 - 1).e^{-x + 3y} what is \frac{\partial^2f}{\partial x \partial y}|_{(-4, -1)} this equation?

Find the second derivative of f(x) = 5x^5 - 4 root 4 of x^7 at x = 3.

Given w = xy + yz + zx, x = 2cos t, y = 2sin t, z = 5t, find dw/dt.

Find the first partial derivatives of f(x, y) = (x - y)/(x + y) at the point (x, y) = (2, 1).

Find the second partial derivative \(partial^2y over partial x^2) of the function. f(x,y) = 3x - x^4 - 4y^2 - 10xy

Find all the second partial derivatives. f(x, y, z) = \frac{3yz}{x + z}

Find all the second partial derivatives. f(x, y) = 3x^2 - xy + 2y^3

Find the partial derivatives: part f / part x and part f / part y. If (a) f(x, y) = xe-2xy (b) f(x, y) = y ^2exy^2

Classify the following second-order linear partial differential equation. 17 u_{y y} + 3 u_{x} + u = 0 a) parabolic b) hyperbolic c) elliptic d) none of these

Verify the linear approximation at (0, 0). (2x + 3)/(4y + 1) approx 3 + 2x - 12y

Find the differential of the function. z = e^(-2x) cos 2 pi t

Find the differential of the function. T = v/(1 + uvw)

Find partial z/partial x and partial z/partial y. x^2 + 8y^2 + 5z^2 = 1.

If \int \frac {dx}{(x + 1)(x - 2)} = A \ln(x + 1) + B \ln (x - 2) + C, then which option is true? a. A + B = 0 b. \frac {A}{B} = -1 c. Both a and b d. None of the above

Find the derivative: y= 2^x (sin x)

Suppose f(x, y) = 5(x - 2)^2 + 4(y - 3)^2. Then nabla f = (0, 0) at the point: A. (5, 4) B. (2, 3) C. (0, 0) D. none of these

Use the equations to find partial z/partial x and partial z/partial y. x^2 + 4 y^2 + 3 z^2 = 1

Find the value of k if the equation is exact: (2 tan x + k sin x sin y) dx + (3 cos x cos y + y) dy = 0.

The volume of a right circular cone is , where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) F...

Find the first partial derivatives of the function. f(x,y) = x over (x + y)^2

Find the gradient of the function

If cos(xyz) = 1 + x2y2 + z2, find

If z = y + f(x2-y2), where f is differentiable, show that

Find all second partial derivatives of f. f(x,y, z) = xkylzm

Find all second partial derivatives of f. z = xe-2y

If z = sin(x + sin t), show that

Find all second partial derivatives of f. f(x, y) = 4x3- xy2.

Find the first partial derivatives.

Determine the first partial derivatives. g(u,v).

Find the first partial derivatives. G(x, y, z) = exzsin(y/z)

Find the first partial derivatives. f(x y) = (5y3 + 2x2y)8

Find all the second partial derivatives. T = e^(-7r) cos(theta).

Given f(x, y) = -x^2 + 2x^2 y^3 + y^4, find A) fx(x, y) B) fy(x, y) C) fxx(x, y) D) fxy(x, y)

Find partial/partial x (3e^(4xy)).

Given f(x, y) = sqrt(3x^2 + 3y^2), find fx(4, 5).

Find all the second partial derivatives of f(x,y) = y tan 2x.

Given f(x, y) = 4x^4 + 6xy^5 + 3y^2, find the following numerical values: A) fx(2, 4) B) fy(2, 4)

Consider the point P(1, 2) and the function given by z = f(x, y) = 2x + 5x^2y - 2y^4. Find the gradient of f.

Let f(x,y) = arctan(xy). Compute the second-order partial derivatives of f.

Find partial^2/partialy partial x (cos(x^2y^2)).

Find the first partial derivatives of the function. w = \ln (x + 6y + 3z)

Find all the first order partial derivatives for the following function. f(x, y, z) = ln(xy)^z.

Find all the second-order partial derivatives of the following function. w = 8x sin(2x^2 y).

Find the four second partial derivatives. Observe that the second mixed partials are equal. z = 21xe^y - 23ye^(-x).

Find all the second-order partial derivatives of the function f(x, y) = 6x^2 + 2y + 7x^2 y^2.

Find f_x(x, y) and f_y(x, y). Then find f_x(2, -1) and f_y(-3, -1). f(x, y) = -8e^(4x - 5y).

Find f_x(x, y) and f_y(x, y). Then, find f_x(-4, -1) and f_y(4, -3). f(x, y) = -4xy + 8y^4 + 1.

Let z = f(x, y) = 20x^2 - 16xy + 25y^2. Find the following using the formal definition of the partial derivative. A. partial z/partial x B. partial z/partial y C. partial f/partial x (5, 3) D. f_y(...

Find the total differential. w = x^13 yz^5 + sin(yz).

Find all the second-order partial derivatives of the following function. w = 8x sin (2x^2y)

Find fx, fy, and fz, and evaluate each at the given point. f(x, y, z) = x^2 y^3 + 2xyz - 5yz, (-3, 1, 3).

Differentiate implicitly to find the first partial derivatives of w. x^2 + y^2 + z^2 - 9yw + 6w^2 = 7

If two resistors with resistances R_1 and R_2 are connected in parallel, then the total resistance, R, measured in ohms, is given by 1/R = 1/R_1+1/R_2 . If R_1 and R_2 are increasing at rates of 0....

Define zx(x, s), when z(x, s) = -ln(9/x - 8 * s) * x^2.

A supermarket sells two brands of granola: brand A at $p per pound and brand B at $q per pound. The daily demands x and y (in pounds) for brands A and B, respectively are given by the following equ...

Suppose we have a function g(m,l) that measures how much enjoyment we get from money m and leisure time l. Describe what information the partial derivative \frac{\partial g (m, l)}{\partial l} prov...

Find the partial derivative. Find f_y (7, 2) when f(x,y) = ye^{xy}. Leave the answer in terms of e. a. 14 e^{14} b. 2e^{14} + 14 e^{14} c. e^{14} + 2 e^{14} d. e^{14} + 14 e^{14}

Use implicit differentiation to find the specified derivative at the given point. Find \frac{\partial y}{\partial x} at the point (1, 3, e^4) for \ln (xz)^y + 6y^3 = 0.

For the given function, F(w,x,y,z)=3xy2+w3z3/32y-xy2z3/w. Evaluate the derivative below. ( / w( / z( F/ x)w,y,z)w,x,y)x,y,z

For the given function, F(w,x,y,z)=3xy2+w3z3/32y-xy2z3/w. Evaluate the derivative below. ( / x( F/ z) w,x,y)w,y,z

For the given function, F(w,x,y,z)=3xy2+w3z3/32y-xy2z3/w. Evaluate the derivative below. ( z( F/ x)w,y,z)w,x,y

For the given function, F(w,x,y,z)=3xy2+w3z3 32y-xy2z3w. Evaluate the derivative below. ( F/ y)w,x,z

For the given function, F(w,x,y,z)=3xy2+w3z3 32y-xy2z3w. Evaluate the derivative below. ( F/ W)x,y,z

For the given function, F(w,x,y,z)=3xy2+w3z3 32y-xy2z3w. Evaluate the derivative below. ( F x)w,y,z

Consider the vector field, F(x, y, z) = (8 e^{sin(y)}, 2 e^{ sin (z)} , 2 e^{ sin(x)} ) a) Find the curl of the vector field. b) Find the divergence of the vector field.

If f(x, y) = 6x^4 - 17xy + 4y^5, then find f_yx.

Given the function f(x, y) = x^2 sin y + y^2 cos x, compute f_x(pi/2, pi).

Find the partial derivative (partial f)/(partial x partial y) where: f(x, y) = e^(x^2 + y^2).

Compute the gradient of the following function f(x, y, z) = e^ 2x sin(2x) cos(2x) .

Find values of x and y such that both f_x(x, y) = 0 and f_y(x, y) = 0. f(x , y) = 3x^2 + 3y^2 - 6x - 6y + 2.

Find f_xy (5, 4) if f(x, y) = (x + 3) ln(xy^4).

Find f_yx (-5, -3) if f(x, y) = 2xe^(3y).

Find f_xy (-4, 3) if f(x, y) = 5x^3 y^2 - 4x^4 - 2xy^3.

Let z=-f(xy) = x^3y-y^4, and let p=(2,1,7) be a point on the surface z=f(xy). Find partial z/partial y

Let z=-f(xy) = x^3y-y^4, and let p=(2,1,7) be a point on the surface z=f(xy). Find partial z/partial x

Determine whether the following function is harmonic or not. u=e^x (x cosy - y sin y)

Find the second partial derivative for the function. f(x,y)=x^3+x^2y^3-2y^2

Find the derivative at x = c (if it exists). f(x) = x^3 + 3x, c = 4

Show that F(x,y,z) = y^2z^3 i + 2xyz^3 j + 3xy^2 z^2 k is a conservative vector field. Then find a function f such that F(x,y,z) = bigtriangledown f.

Considering the following: zy + x\:ln(y) = z^2. Find \delta z/ \delta x.

Find a function w = f(x, y) whose first partial derivatives are f_x = 1 + e^x cos y and f_y = 2 y - e^x sin y and whose value at the point (ln 2, 0) is ln 2.

If f(x, y) = (y)^2 x. Verify f_xy = f_yx.

Find the rate of change of the function f(x, y) = 3 sin (x) cos (y) - y at the point (pi/2, pi) in the direction of the vector u = langle 3, 4 rangle.

Let \displaystyle{ f(x, y) = xe^x + x^2y +y^2 . } Determine \displaystyle{ \dfrac{ \partial^2 f}{\partial x^2}, \ \dfrac{ \partial^2 f}{\partial y^2}, \ \dfrac{ \partial^2 f}{\par...

Let \displaystyle{ f(x, y) = x^2y + 2xy^2 . } Determine \displaystyle{ \dfrac{ \partial^2 f}{\partial x^2}, \ \dfrac{ \partial^2 f}{\partial y^2}, \ \dfrac{ \partial^2 f}{\partial...

Let \displaystyle{ f(x, y) = xy^2 + 5 . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x} \text{ and } \dfrac{ \partial f}{\partial y} \text{ at } (x, y) = (2, 1). }

Let \displaystyle{ f(x, y) = (x + y^5)^5 . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x} \text{ and } \dfrac{ \partial f}{\partial y} \text{ at } (x, y) = (1, 2). }

Let \displaystyle{ f(x, y) = x^2 + 2xy y^2 + 6x +6y . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x} \text{ and } \dfrac{ \partial f}{\partial y} \text{ at } (x, y)...

Let \displaystyle{ f(x, y, z) = \dfrac{ xy}{z} . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x}, \ \dfrac{ \partial f}{\partial y}, \text{ and } \dfrac{ \partial f}{\...

Let \displaystyle{ f(x, y, z) = ze^{x/y} . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x}, \ \dfrac{ \partial f}{\partial y}, \text{ and } \dfrac{ \partial f}{\partial...

Let \displaystyle{ f(x, y, z) = (1 + z^2 y) \sqrt{x} . } Determine \displaystyle{ \dfrac{ \partial f}{\partial x}, \ \dfrac{ \partial f}{\partial y}, \text{ and } \dfrac{ \partial...

Let \displaystyle{ f(p, q) = 1 p(1 + q) . } Determine \displaystyle{ \dfrac{ \partial f(p, q)}{\partial p} \quad \text{and} \quad \dfrac{ \partial f(p, q)}{\partial q}. }

Let \displaystyle{ f(L, K) = 3 \sqrt{LK} . } Determine \displaystyle{ \dfrac{ \partial f(L,K)}{\partial L} \quad \text{and} \quad \dfrac{ \partial f(L,K)}{\partial K}. }

Let \displaystyle{ f(x, y) = \sqrt{x^2 + y^2} . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Let \displaystyle{ f(x, y) = \ln(xy) . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Let \displaystyle{ f(x, y) = x^5e^{2y} \ln(y) . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Let \displaystyle{ f(x, y) = \dfrac{ 5^x}{1 + e^y} . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Let \displaystyle{ f(x, y) = \left( 2x y + 5 \right)^{2/3} . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\pa...

Let \displaystyle{ f(x, y) = \dfrac{ x}{y} + \dfrac{ y}{x} . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\par...

Let \displaystyle{ f(x, y) = x^2 y^2 . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Let \displaystyle{ f(x, y) = 5xy . } Determine \displaystyle{ \dfrac{ \partial f(x,y)}{\partial x} \quad \text{and} \quad \dfrac{ \partial f(x,y)}{\partial y}. }

Find all first-order partial derivatives: R(x, y) = x^2 / y^2 + 1 - y^2 / x^2 + y

Find the critical point of the function and then determine whether it is a maximum, local minimum, or saddle point. f(x,y) = x^2+y^2+8xy

Find the critical point of the function and then determine whether it is a maximum, local minimum, or saddle point. f(x,y) = x^2-y^2+3xy

The temperature at any point (x, y) in a steel plate is T = 700-0.7x^2 -1.3y^2, where x and y are measured in meters. At the point (8, 8), find the rates of change of the temperature with respect t...

For each of the following vector fields F, decide whether it s conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vf F with f...

Consider the surface F(x, y, z) = x^6z^6 + sin(y^3z^6) +1 = 0. . Find the following partial derivatives: (a) dfrac{ partial z}{ partial x} (b) dfrac{ partial z}{ partial y}

Calculate Jacobian for spherical coordinates. x = rho sin phi cos theta y = rho sin phi sin theta z = rho cos phi

Let f(x, y) = x^2 + 2 x y - 4 y^2 + 4 x - 6 y + 4. Use this function to do the following: (a) Calculate Delta f. (b) Calculate the directional derivative of f at (1, -1) in the direction of v = 2 i...

Use the level curves of the function z = f(x. y ) to decide the sign (positive, negative, or zero) of each of the partial derivatives at the point P. Assume the x and y axes are in the usual positi...

Find the partial derivatives, when w = 4x^2+4y^2+z^2,x=row sin phi cos theta , y= row sin phi sin theta ,z= row cos theta. Find.

Consider the initial value problem partial^2 u/partial x^2 = partial=^2 u/partial t^2, where - infinity less than x less than infinity, t greater than equal to 0. If u(x,0) = Square root of{x} and...

Consider the Laplace equation partial^2 u/partial x^2 + partial^2 u/partial y^2=0, where 0 less than equal to x less than equal to 4 and 0 less than equal to y less than equal to 4. Given boundary...

Consider the initial value problem partial^2 u/partial x^2=1/9 partial^2 u/partial t^2, where -infinity less than x less than infinity, t greater than equal to 0. If u(x,0) = 3 and u_t(x,0)=e^3x,...

Find all the second-order partial derivatives of the following function w = 4 x^2 tan (3 x^7 y).

If w = x^2 + y^2, where x = r - s and y = r + s, then \frac(\partial w)(\partial r) = a. 2r + s. b. r + s. c. 4s. d. rs. e. 4r.

Find the increment and differential of the function f = exsin(y-z); x = y = z = 1, dx = -0.2, dy = 0.3,dz = 0.02.

If w = x2 + y2, where x = r-s and y = r+s, then partial w/ partial r is 1. 2r+s 2. r+s 3. rs 4. 4r 5. 4s

Show F(x, y) = (2xy3 +9)i +(3x2y2 +2e(2y))j is conservative by finding a potential function f for F, and use to compute F. dr, where C is the curve given by r(t) = 2 sin9ti + (2t/pi) sin 8(5t)j for...

If w = x^2 + y^2, where x = r - s and y = r + s, then partial w / partial r = ____.

If w = x^2 + y^2, where x = r - s and y = r + s, then find \dfrac{\partial w}{\partial r}.

If w=x^2+y^2, where x=r-s and y=r+s, then find partial w/partial r.

Find the Hessian matrix and its determinant (Hessian, henceforth) evaluated at the point (0,0). f(x,y) = 1/e^{(x^2+y^2)}

Find a potential function for the vector field. F(x,y)= (3x+y,x2 - 3y).

If f(x, y) = cube root(x3 + y3). Find fx (0,7).

x2 + 2y2 + 5z2 = 1 Find

Calculate all first and second partial derivatives f (x, y, z) = x y e^x^2z.

If z is implicitly defined as a function of x and y by \displaystyle{ x^2 + y^2 + z^2 = 1, } show that \displaystyle{ x \dfrac{ \partial z}{\partial x} + y \dfrac{...

Determine the critical points of the following function and find out whether each point corresponds to a relative minimum, maximum, saddle point or no conclusion can be made: f(x,y)= x^2 + 3y^2 - 2...

Let z=f(x,y) be a function defined implicitly in the variables x and y such that: determine partial z/partial x(0,2).

Classify whether the following (i) parabolic (ii) hyperbolic (iii) elliptic. (a) U_{xx}+U_{xy}-U_{yy}-U_x-U_y=0 (b) U_{xx}+U_{xt}+3U_{tt}+U_x+U_t=0

Given f(x, y) = x^4 + 5 xy^3, find all of the first and second partial derivatives.

What is the first order partial derivative of f ( x , y ) = \frac { x - y } { x + y } with respect to x?

Find the second partial derivatives. a) f(x, y) = sin(x-y)/e(-xy) b) g(x, y) = ln(sqrt(x)+ 2(y)

Find the linearization of the function f(x, y) = square root 18 - 2 x^2 - 3 y^2 at the point (1, -2).

Find . f(x,y) = 1000+4x-4y Evaluate them all at (1,-1) if possible.

What is the degree of the partial differential equation fraction partial ^2 z partial x^2 + fraction partial z partial y ^3 + z = tan xy.

Find all second-order partial derivatives of the function f_x, y (x, y) = cos(2 x y^2 - 3x^2y^3).

Compute all the first and second partial derivatives f x, f t, f xx, f tt, f tx of the function f x, t = x cos 3t^3 - 2xt.

Let w = 3 x^2 y - e^2 x y with x = s + t cos (r), y = s + t sin (r), and z = s t r^2. Find partial w / partial t at the point (r, s, t) = (pi / 3, 2, 1).

Calculate that if \displaystyle{ \omega = \left( \dfrac{ x y + z }{x + y z} \right)^h } then \displaystyle{ x \dfrac{ \partial \omega}{\partial x} + y \dfrac{ \partial \omega}...

Find the first-order partial derivatives of f(x,y) = 9x-11y 6x + 5y.

Consider f(x, y) = 4x^4 + y^2 - 2xy + 1, evaluate D = f_{xx} f_{yy} - (f_{xy})^2 at (0, 0).

Find f_y for the function f(x,y) = x+y/(xy-1).

Find f_x for the function f(x,y) = x+y/(xy-1).

The equation 2z^3=ln(e^{2z}+2xy) implicitly defines z as a function of x and y in the neighborhood of the point where x=2, y=0, and z=1. Find partial z/partial y at this point.

The equation 2z^3=ln(e^{2z}+2xy) implicitly defines z as a function of x and y in the neighborhood of the point where x=2, y=0, and z=1. Find partial z/partial x at this point.

Evaluate the indefinite integral. integral eighth root of cot (x) csc^2 (x) dx

Use the definition of the partial derivative to find f y fx, y = x^3y^2 + 4y

If f(x, y) = tan^-1 (2 x / 3 y), verify f_x y = f_y x.

Calculate all four fxx x, y, f xy x, y, f yx x, y, f yy x, y second-order partial derivatives of f x, y = 4x^2y + 5xy^3.

Find the indicated partial derivatives of f(x, y) = 2 x^3 y^2 using the limit definition. The limits need to be reduced as much as possible before they are evaluated.

For the function: f(x,y) = 8x2y-8y4 +ysin(xy), find f(xy)(x,y).

Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x, y)=ln...

Consider the function f(x, y) = x^2 + y^2 - 3xy + 25x. Use the Second Derivative Test to determine what statement is true. A) Critical point is a local minimum B) Critical point is a saddle point...

Find the 4 second-order partial derivatives of the function f(x, y) = \frac{2y}{x + y}.

For the function f(x,y) = x^2e^{7xy}, find f_y(-2,5).

For the function f(x,y) = x^2e^{7xy}, find f_x(5,-1).

For the function f(x,y) = x^2e^{7xy}, find f_y.

For the function f(x,y) = x^2e^{7xy}, find f_x.

For f(x, y) = e^y sin x, evaluate f_x at the point (pi, 0).

Find f_x(x, y) and f_y(x, y). Then find f_x (2, -1) and f_y(-3, -3). f(x, y) = - 7 e^-4 x - 3 y

Compute all the first and second partial derivatives of the function f (x, t) = x cos (2 t^4 - 2 x t).

Find u_xx'' when u and v are defined as functions of x and y by the equations x y + u v = 1 and x u + y v = 0.

Compute all second-order partial derivatives of the following function: f x, y = x + 2y ^2 - e^ x^2y

Compute all the first and second partial derivatives of the function f (x, t) = x + 3 / x t - 1.

For the function given below, find the numbers x, y such that f x x, y = 0 and f y x, y = 0. f x, y = 5x^2 + 19y^2 + 19 xy + 7x - 3

Find the partial derivatives f x x, y, z, f y x, y, z, f z x, y, z, f yz x, y, z of the function f x, y, z = x^6 + 4yz^2 + z^4

Find all the first order partial derivatives fx x, y, f y x, y and second order partial derivatives f xx x, y, f xy x, y, f yy x, y of the function f x, y = 4ye^ 5x

Find the partial derivatives of the function f (x, y) = -8 x y + 4 y^3 + 2. a. f_x(x, y). b. f_y(x, y).

Find all the first order partial derivatives fx x, y, f y x, y and second order partial derivatives f xx x, y, f xy x, y, f yy x, y of the function f x, y = 5x^2y^2 - 2 x^2 + 7y

Find the partial derivatives f x x, y and f y x, y of the function f x, y= fraction x^5 y^3 x^2 + y^2.

Find the partial derivatives f x x, y and f y x, y of the function f x, y = x sin 2x^3y.

Find the partial derivatives f x x, y and f y x, y of the function f x, y = e^ 5x - 3y

Find the partial derivatives f x x, y and f y x, y of the function f x, y = -6x^4 y^3 + 5.

Find the partial derivatives f x x, y and f y x, y of the function f x, y = -8xy + 4y^3 + 2.

Consider the function f(x,y)= x^4+y^4-4 xy+2. Compute the second-order partial derivatives.

Consider the function f(x,y)= x^4+y^4-4 xy+2. Compute the first-order partial derivatives.

Find all first and second-order partial derivatives of the following function. y = 4x4z3 - z2 + zx + 10z2x + z - x

Find the Jacobian of the Transformation for x=u^2+uv, y=9uv^2.

Find the Jacobian of the Transformation for x=2u+v, y=4u-v.

Compute two partial derivatives (choose two of partial-x, partial-y, partial-z) for the following function: f(x, y, z) = e^(xy - 1) ln(x^2 - cos(z)).

Suppose f: R^2 to R satisfies sin(x f(x,y)) - xy = x^2 for all points (x, y) and f(2, -2) = 0. Find the partial derivatives of f at (2, -2).

A company produces two types of solar panels per year: x thousand of type A and y thousand of type B. The revenue and cost equations, in millions of dollars, for the year are given as follows. R(x....

Given f(x, y) = sin square root x^2 + y^2. Assume the given function is continuous and all partial derivatives exist. Find the second partial derivatives partial^2 f / partial x^2, partial^2 f / pa...

Find the first order partial derivatives z_x(x, y) and z_y(x, y), where z(x, y) is defined by x^3y^2z = sin(z(x^2 - y^2)).

Find all the critical points of the following function and determine their nature. f(x,y)=e(x +3y2)

For the functions f x, y, z = - 8e^xyz , g t = 8t, - 5t^2, t^3 find nabla f , g' and f circ g' 1.

Find f_x and f_y if \displaystyle{ f(x, y) = (6x + 8y)(6x 8y). }

Consider the function f(x,y) = 2x3 + 6xy - 343y - 150x. Find all the saddle points.

Consider the function f(x,y) = 2x3 + 6xy - 343y - 150x. Find all the local minima.

Consider the function f(x,y) = 2x3 + 6xy - 343y - 150x. Find all the local maxima.

Let f: real{R}^3 to real{R} be differentiable, and let c: (0,1) to real{R}^3 be contained in a level set of f. Show that c'(t) is perpendicular to bigtriangledown f (c(t)) for all t.

Find the partial differential equation by eliminating the arbitrary constants from z = axy

Solve 4 fraction partial u partial x + fraction partial u partial y = 3u and u 0, y = e^ - 5y by the method of separation of variables

Determine the value of k so that the given differential equation is exact. Then use it (the value of k) to solve this equation. (y^2 / x + cos y) dx + (k y ln x - x sin y + y e^y) dy = 0

Given \displaystyle{ A(x, y) = \cos \left( \dfrac{ x}{y} \right) x^7y^4 + y^{10}. } Is A_{xy} = A_{yx} ? Show.

Consider the function f(x,y) = 3(2x - y)2. Find the saddle points?

Find the parametric equations of the normal line to z = x^3y^2 + x / y at (x, y, z) = (2, -1, 6).

Given that f(x,y) is differential at (3,8) with: lim_{(x,y) to (3,8)} = 7, f_x (3,8)=-4 and f_y=(3,8)=6. Use partial differentiation to approximate the quantity: A=f(3.004, 7.9).

For the function f(x, y, z)= x^2 yz^3 + e^x + y + 2 z. Evaluate partial f / partial z|_(3, 3, -3).

Find the indicated partial derivatives of the function f(x, y) = x^3 e^4 x y + x cos y. Write only the final answer. a) partial f / partial x. b) partial f / partial y. c) partial^2 f / partial x p...

Find the general solution of U_xx + 2 U_x t - 3 U_tt = 0.

Let f x, y = x^2y + fraction 1 3 y^3. The f xx + f yy is a. 4y b.2x + 2y c. 4x d. 2xy

Find the parametric equations of the normal line to the function f (x, y) = x^2 + 3 x y + y^2 at the point (1, -3).

Given z = z(x, y) implicitly as \displaystyle{ \ln(xz) + y \ln(x) x^2 + 4 = 0. } Find \displaystyle{ \begin{alignat}{3} \dfrac{ \partial z}{\partial x} \text{ and } \dfrac{ \...

Compute f_x (1, 0) and f_y(2, - 1) of the function f(x, y) = 3 x^2 + x y + 4 y^3.

Find the equation of the plane tangent to f(x, y) = sin(2x + 7 y) + 7 at the point (7, -2).

Find the parametric equations of the normal line to the function f(x, y) = x^2 + 4xy + y^2 at the point (2, -3).

Consider the function, f(x,y) = (xy (x2 - y2)) / (x2 + y2), (x,y) not equal to (0,0), 0, (x,y) = (0,0). Show that fx(0, 0) = 0 and fy(0, 0) =0.

If z = x^2 - x y + 8 y^2 and (x, y) changes from (3, -1) to (2.96, -1.05), compare the values of Delta z and dz. (Round your answers to four decimal places.)

Consider the function f x = x - 1 square root 3 x. A. Find the relative extrema of f. B. Find all inflection points of f. C. Sketch the graph of f.

f(x,y) = xy^2 + ye^x^2 + 5. Find partial differentials:

Find partial f / partial x, partial f / partial y, and partial f / partial z. f (x, y, z) = 21 e^ x y z

Consider f(x, y) = y ln (3 x + 2 y). Find f_x(x, y).

Given the function \displaystyle{ f(x,y) = x^2y^3 2e^x y + 4yx^2, } show that f_{xy} = f_{yx}.

Find f_z y x for f(x, y, z) = x^5 y^6 z^5 + x y^9 + y^6 z.

Consider the following: g(x,y) = \dfrac{xy}{3x - 4y}. Find \dfrac{ \partial^2 g}{\partial x \; \partial y}.

Consider the following: f(x,y) = e^{3xy-4x^2y^2}, \: (2,1). Find f_y(2, 1).

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