Find {eq}\displaystyle \frac{\partial z}{\partial y} \ {/eq} if {eq}\ xyz = cos(2x + 3y -z) {/eq}.

## Question:

Find {eq}\displaystyle \frac{\partial z}{\partial y} \ {/eq} if {eq}\ xyz = cos(2x + 3y -z) {/eq}.

## Partial Derivatives:

When taking a partial derivative with respect to, {eq}x {/eq}, you treat the variable {eq}y {/eq} as if it is a constant. When we take the partial derivative with respect to, for example, {eq}x {/eq}, we take all other variables as constant for that particular instant. Partial derivatives are used in a variety of applications, including Lagrange multipliers, absolute maxima and minima, gradient vectors, tangent and normal lines, and many others.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answerGiven {eq}\displaystyle xyz = \cos(2x + 3y -z) {/eq}

Partially differentiating {eq}\displaystyle z {/eq} with respect to {eq}\displaystyle ...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 18 / Lesson 12What is a Partial Derivative? Learn to define first and second order partial derivatives. Learn the rules and formula for partial derivatives. See examples.

#### Related to this Question

- Find partial z over partial x and partial z over partial y by implicit differentiation if ye^x - 5 sin 3z = 3z.
- If z = u^2-v^3, u=e^{2x-3y}, v=sin (x^2-y^2), find partial z / partial x and partial z/partial y
- Find partial z / partial x + partial z / partial y, if sin (2 x y z) = x^3 + y^3 + z^3.
- Find partial z / partial x if 1 / x + 1 / y + 1 / z = 0.
- Find {partial z / partial x} if z = (x^2 + 7 x y + y^2)^8.
- Determine {partial z} / {partial x} for x^2 - (y + z)^2 = 0.
- If z = sin (x^2 y), x = ln (s t^2), y = s^2 + 1 / t. Find {partial z} / {partial s} and {partial z} / {partial t} derivatives.
- Find {partial z} / {partial x} and {partial z} / {partial y} if e^{-x y} - 3 z + e^z = 0.
- Find partial z / partial x and partial z / partial y, if x y z = tan (x + y+ z).
- Find partial z / partial x for 3 x z^2 - e^4x cos 4z - 3 y^2 = 4. Do not simplify.
- Find partial z / partial x and partial z / partial y. x^2 - y^2 + z^2 - 2z = 4
- Find partial z / partial x and partial z / partial y. e^z = xyz
- Find partial z/ partial x and when partial z/ partial y when z is implicitly given as a function of x and y by the equation: a. x^2 + y^2 + z^2 = 3xyz b. yz = In(x + z)
- Find partial^2 z/partial y partial x for z=4x^2+y^7.
- Find partial^2 z/partial x partial y for z=4x^2+y^7.
- Find {partial z} / {partial x} given the following. z = 3 x y z - x^6 y^5 + e^{-16 x y^2}
- If z = x^2y - y^2\sin^{-1} x, find partial z/partial y at (1, 2).
- Let z = 4 x^3 cos (3 x y). Find {partial z} / {partial x}.
- For the function z = f ( x , y ) = - x ^ { 3 } + 9 y ^ { 2 } - 3 x y , find \displaystyle\frac { \partial z } { \partial x } , \frac { \partial z } { \partial y } , f _ { x } ( - 1 , - 1 ) , and f _
- Use the equations to find partial z / partial x and partial z / partial y. x^2 + 8 y^2 + 7 z^2 = 1.
- Suppose z = xye^z + x -4. Then find \partial z/ \partial y at (x,\ y,\ z) = (2,\ 1,\ 0).
- Find partial z / partial x and partial z / partial y of the following function. z = (x^2 e^ {3 y} + y^2 e^ {3 x}).
- Find partial z/partial v when u = 0 and v = 11pi/2 if z(x, y) = sin x + cos y, x = u dot v, and y = u + v.
- Given z = x^4 + xy^2, x = uv^4 + w^3, y = u + ve^w then find: partial z/partial v when u = -2, v = -3, w = 0.
- Determine partial z/partial x and partial z/partial y if z is defined implicitly as a function of x and y by the equation x^8 + y^6 + z^6 = 7xyz.
- If w = x / 2 y + z then find partial^3 w / partial y partial z partial x and partial^3 w / partial x^2 partial y.
- Suppose z = x2 sin y, x = 4s2 + 5t2, y = -6st. Use the chain rule to find partial z/ partial s and partial z/ partial t functions of x, y, s and t.
- Let z = {x y} / {4 y^2 - 4 x^2}. Then, find: (a) {partial z} / {partial x} (b) {partial z} / {partial y}
- Let z = 3e^x^2y^4, then find partial z/partial y and partial z/partial x.
- Suppose z = x^2 sin y, x = -2s^2 - 2t^2, y = 0st. A) Use the chain rule to find partial z/partial s and partial z/partial t as functions of x, y, s and t. B) Find the numerical values of partial z/partial s and partial z/partial t when (s, t) = (-1, -1).
- Find: a) \partial z/ \partial x, if 3x^2 z + y^3 - xyz^3 = 0; b) \partial x/ \partial z, if ye^{-x} + z sin x = 0
- If z = x y^2 + x^2 y where x = 3 u^2 - 2 v + 1 and y = 2 u + v^3, find {partial z} / {partial u} and {partial z} / {partial v}.
- Suppose z = x^2 sin y, x = 1s^2 + 2t^2, y = -2st. A. Use the chain rule to find partial z/partial s and partial z/partial t as functions of x, y, s, and t. B. Find the numerical values of partial z/partial s and partial z/partial t when (s, t) = (3, 3).
- Suppose z = x2 sin y, x = 1s2 + 2t2, y = -2st. (a) Use the chain rule to find partial z/partial s and partial z/partial t as functions of x y, s and t. (b) Find the numerical values of partial z/partial s and partial z/partial t when (s, t) = (3,3).
- Find partial z / partial y for xy^2 + ln(x^2y) + z squareroot x - squareroot 3 = 0 by performing the implicit differentiation.
- Suppose z = x^2 sin y, x = 3 s^2 + 3 t^2, y = 2 s t. A. Use the Chain Rule to find partial z / partial s and partial z / partial t as functions of x, y, s and t. partial z / partial s = partial z /
- Find partial z / partial y at (1/2, - 2, 2), where x and y are independent of each other, and z e^{x y + 1} = [ (sin ((pi x y) / 6) / (1 + x + y)) ] + 3
- If cos (x y z) = 1 + x^2 y^2 + z^2, find partial z / partial x and partial z / partial y.
- Suppose z = x2 sin y, x = -5s2 + 2t2, y = -10st. (a) Use the chain rule to find partial z/ partial s and partial z/ partial t as functions of x, y, s, and t. (b) Find the numerical values of partial z/ partial s and partial z/ partial t when (s, t) = (3,
- Find the indicated partial derivative(s). w = x / y + 2 z a. partial^3 w / partial z partial y partial x. b. partial^3 w / partial x^2 partial y.
- 1. Given that (x^2)y + (x^2(y^3) - xz + zy(^2) = 0, find the partial of z with respect to x and the partial of z with respect to y. 2. Let f(x,y,z) = (sin^-1)(z/x) + (y^2)z. Calculate the partial
- Let z = e^{2r}sin(3 \theta) \theta = \sqrt{s^{2} + t^{2 r = st - t^{2} Find \partial z/\partial t.
- Determine partial z / partial x and partial z / partial y by differentiating implicitly. (a) 3 x^2 + 4 y^2 + 2 z^2 = 5. (b) square root x + y^2 + sin (x z) = 2.
- Find the partial of z with respect to x and the partial of z with respect to y at the point (0,0,0) for the equation: sin(-2x-4y+z)=0.
- Suppose that z = e x 2 y where x = ? u v and y = 1 v . Find partial z/ partial u and partial z/ partial v in terms of u and v . Simplify your answers
- Find {partial z}/{ partial x}, { partial z}/{ partial t} where z = x^y + x, x = sqrt{s + t} ; and ; y = s^2t
- Suppose that z is implicitly defined by the equation: y z^2 + x^2 ln y = tan z. Find partial z / partial x and partial z / partial y.
- Find partial^2 z/partial x partial y for z=y^3-6xy^2-3.
- Find partial^2 z/partial y partial x for z=y^3-6xy^2-3.
- If z = sin(x) cos(y), where x = uv^2 and y = u^2v, find \partial z /\partial u.
- Suppose z = x^{2}\sin y, x=-2s^{2}+0t^{2}, y=-6st.Use the chain rule to find \frac{\partial z }{\partial s} and \frac{\partial z }{\partial t} as function of x, y, s and t.
- If sin(4x + 4y + z) = 0, find the first partial derivatives fraction {partial z}{partial x} and fraction {partial z}{partial y} at the point (0, 0, 0).
- For the equation x^3 + y^4 + z^5 = 4xyz, find the partial of z with respect to x and the partial of z with respect to y.
- Consider the following equation sqrt(x^2 + y^2 + 2z^2) = cos z. (i) Can we solve for y as functions of x, z near (x, y, z) = (0, 1, 0)? If so, find \partial y/ \partial x and \partial y/\partial z a
- If x y z + z^2 = 15 defines z implicitly as a function of x and y, then find {partial z} / {partial x}|_{(2, 1, 3)}.
- If z=(x+y)e^{x+y}, x=u, y= ln (v), find partial z/partial u and partial z/partial v.
- Calculate partial z / partial x and partial z / partial y at the points (3, 2, 1) and (3, 2, -1), where z is defined implicitly by the equation z^4 + z^2 x^2 - y - 8 = 0.
- Find partial^2 z/partial x^2 for z= square root{81-x^2-y^2}
- Suppose z = x2 sin y, x = 4s2 + 5t2, y = -6st. Find the numerical values of partial z/ partial s and partial z/ partial t when (s, t) = (2, 2).
- Find partial z/ partial u and partial z/ partial v where z = f(x, y) = x tan (x/y) and x = uv, y = u/v; expressing your answers in terms of u and v only.
- If z = xe^y, x = u^3+v^3, y = u^3-v^3, find the partial derivative partial z/partial u and partial z/partial v. The variables are restricted to domains on which the functions are defined.
- Consider f (x, y, z) = x / {y - z}. Compute the partial derivative below. {partial f} / {partial x}_{(2, -1, 3)}.
- 1. z = x^3y^5. x = rs^2. y = r^2 + s. Determine the value of partial z/partial s when r = 1 and s = -2. 2. z = xy^2 + x cos y. Determine the values of partial^2z/partial x^2, partial^2z/partial y^2,
- Suppose z = x y e^x + x - 4. Then {partial z} / {partial y} at (x, y, z) = (2, 1, 0) equals: A) 2. B) 1/2. C) -1 /2. D) -2.
- Find the first partial derivatives of the function. z = (3 x + 8 y)^5 partial z/partial x = partial z/partial y =
- Z = cos (x^2 + y^2), x = u cos (v), x = u cos(v), y = u sin(v), find partial difference \frac{\partial z}{\partial u} and\frac{\partial z}{\partial v}.
- Suppose z = x^2 sin y, x = 4s^2 - 4t^2, y = 10st. Find the numerical values of partial derivatives of z in terms of s and t when (s, t) = (5, -4). Provide a step-by-step solution to your answer.
- Use the following equation to find \frac{partial z}{partial y} \frac{partial z}{partial x}=- \frac{\frac{partial f}{partial x}{\frac{partial f}{partial z \frac{ partial z}{ partial y}= - \frac{\fra
- Find (partial^3 u)/(partial x partial y partial z) and (partial^3 u)/(partial x partial y^2) of u = (3x^2)/(2y + z^3).
- Find fraction {partial z}{partial x} if z = (x^2 + 9xy + y^2)^4
- Given 3z^3-4xy + 4yz + y^3 - 6 = 0 , find \frac{\partial z }{\partial x} and \frac{\partial z }{\partial y} at the point (2,4,1)
- Suppose z is given implicitly as a function of x and y by and (a)\vec{F}(x,y,z)=x^{5}+y^{5}+z^{5}+5xyz=0 (b)\frac{\partial z }{\partial x} (c)\frac{\partial z}{\partial y}
- Find the value of partial z / partial x at the point (1, 1, 1) if the equation 8 x y + z^3 x - 4 y z = 5 defines z as a function of the two independent variables x and y and the partial derivative exists.
- Find the value of fraction partial z partial x at the point 1, 1, 1 if the equation 3xy + z^4 x - 3yz = 1 defines z as a function of the two independent variables x and y, and the partial derivative exists.
- Find the value of fraction partial z partial x at the point 1, 1, 1 if the equation 7xy +z^3x - 2yz = 6 defines z as a function of the two independent variables x, y, and the partial derivative exists.
- Find fraction partial z partial u when u = 2, v = 0, if z = sin xy + x sin y, x = 3u^2 + 2v^2, and y = 2uv
- If z = x e^{y} , x = u^{2} + v^{2}, y = u^{2} - v^{2}, find partial differentiation z / partial differentiation u and partial differentiation z / partial differentiation v. The variables are restricted to domains on which the functions are defined.
- Let : z^4 = 8xe^{y/z} calculate the partial derivatives partial z / partial x , partial x / partial z using implicit differentiation ? (a) partial z / partial x = ? (b) partial x / partial z = ?
- Given z = 3x - x^2y^2, solve for \frac{\partial z}{\partial x} and \frac{\partial z}{\partial y} using partial derivatives.
- Find fraction partial z partial x and fraction partial z partial y. z = x^8 ln 1 + xy^- fraction 5 7
- Given: z= x^4 + x y^2, x = uv^4 + w^5, y = u + ve^w Find {partial z} / {partial u} when u = -2, v = 2, w = 0.
- Suppose z = x^2 sin y, x = 3s^2 + 3t^2, y = 6st. A. Use the chain rule to find fraction partial z partial s and fraction partial z partial t as functions of x, y, s, and t. B. Find the numerical values of fraction partial z partial s and fraction partial
- Find the value of fraction partial z partial x at the point 1, 1, 1 if the equation 9xy + z^3 x-2yz = 8 defines z as a function of the two independent variables x and y and the partial derivative exists.
- 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
- Suppose that z is defined implicitly as a function of x \enspace and \enspace y by the equation F(x,y,z) = xz^2+y^2z+ 3xy - 1 = 0 . a. Calculate F_x, F_y, F_z b. Calculate the partial deriv
- Suppose that a function z = f(x, y) is defined implicitly by constraint (x^2 + y^2 + z^2)^2 = x - y + z. Use implicit differentiation to calculate partial z / partial x and partial z / partial y in te
- Find the value of (partial x)/(partial z) at the point (1, -1, -3) if the equation xz + y*ln x - x^2 + 4 = 0 defines x as a function of the two independent variables y and z and the partial derivative exists.
- Suppose that z is defined implicitly as a function of x and y by the equation F(x,y,z)=xz^2 +y^2z+3xy-1=0. (a) Calculate Fx,Fy,Fz. (b) Calculate the partial derivatives \displaystyle \frac{\partial z}
- Let z=f(x,y), x=x(u, v), y=y(u,v) and x(2, 3) = 6, y(2, 3) =5, calculate the partial derivative in terms of some of the numbers a, b, c, d, m, n, p, q. f_x(2, 3)=a f_y(2, 3)=c x_u(2
- Use the chain rule to find expressions for partial z by partial u and partial z by partial w if z equals f of x, y, x equals h of u, w, y equals g of w,t. (Just in case, there are no typos in the abov
- Given that z = e^{xy}; x = 2u + v, y = \frac{u}{v}. find \frac{ \partial z}{\partial u} and \frac{ \partial z}{\partial v} using the appropriate form of the chain rule. Express the partial derivative
- 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.
- Use the Chain Rule to find \partial z/ \partial s and \partial z/ \partial t . (Enter your answer only in terms of s and t .) z = e^r \cos(\theta), r = st, \theta = \sqrt{s^2+t^2} \partial
- For z=(x^3)+x(y^4), x= u(v^4)+(w^3), and y = u+v(e^w), use the Chain Rule to find the partial of z with respect to u, the partial of z with respect to v, and the partial of z with respect to w when u=
- Given that xyz=cos(x+y+z), use the equations below to find the partial derivative of z with respect to x and the partial derivative of z with respect to y. The partial derivative of z with respect t
- If u = 4x^3 + 3y^3 + x^2y+ xy^2, find \partial u/ \partial x and \partial u/ \partial y.
- Let f(x, y) = x^4y^2 - x. Find : a) \partial f/ \partial x b) \partial f/ \partial y\partial x c)\partial^2f/ \partial y^2
- Suppose that z=f\left ( x,y \right ) satisfies xe^{z}+ze^{y}=x+y. Calculate \partial z/\partial x as a function of x, y, and z.
- Given the equation z= ax ln y - y/bz where a and b are constants: find (partial x/partil y)_z, (partial y/partil z)_x, and (partial z/partil x)_y.
- Given z = 2x^3 ln y + xy^2, find frac{partial^2z}{partial y partial x}.