If {eq}z = x^2y - y^2\sin^{-1} x {/eq}, find {eq}\dfrac {\partial z}{\partial y} {/eq} at {eq}(1, 2) {/eq}.

## Question:

If {eq}z = x^2y - y^2\sin^{-1} x {/eq}, find {eq}\dfrac {\partial z}{\partial y} {/eq} at {eq}(1, 2) {/eq}.

## Partial derivatives:

Partial derivative to a function containing more than one variable is the derivative of that function with respect to a single variable, i.e., Other variables are treated as a constant that do not vary.

## Answer and Explanation: 1

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

View this answer

We have,

$$z = x^2y - y^2\sin^{-1} x $$

Differentiate t{eq}z {/eq} with respect to {eq}y {/eq}, treat the variable {eq}x {/eq} as a constant,

...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 14 / Lesson 4This lesson defines the chain rule. It goes on to explore the chain rule with partial derivatives and integrals of partial derivatives.

#### Related to this Question

- 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 = ^{u - 2v}, and u = xy^2, v = x^{2}y, find \frac{\partial z }{\partial x} and \frac{\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.
- Find partial z / partial x for 3 x z^2 - e^4x cos 4z - 3 y^2 = 4. Do not simplify.
- Use implicit differentiation to find partial z partial x and partial z partial y e 2 z = x y z
- 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
- Use implicit differentiation to find partial z / partial x and partial z / partial y. x^5 + y^4 + z^8 = 8 x y z.
- Use implicit differentiation to find partial z/ partial x and partial z/partial y. x^2 + 2y^2 + 7z^2 = 8
- Using implicit differentiation, find (\partial z/ \partial x) and (\partial z/ \partial y), given that y^{2} - 2xz^{2} + xyz - 3 = 0
- Use implicit differentiation to find partial z/ partial x and partial z/ partial y. x^2 + 4y^2 + 5z^2 = 2
- Find partial z / partial y for xy^2 + ln(x^2y) + z squareroot x - squareroot 3 = 0 by performing the implicit differentiation.
- Let z = {x y} / {4 y^2 - 4 x^2}. Then, find: (a) {partial z} / {partial x} (b) {partial z} / {partial y}
- 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 z over partial x and partial z over partial y by implicit differentiation if ye^x - 5 sin 3z = 3z.
- Determine {partial z} / {partial x} for x^2 - (y + z)^2 = 0.
- Let x y z + \frac { 1 } { x y z } = z ^ { 2 }. Use implicit partial differentiation to find \frac { \partial z } { \partial x } and \frac { \partial z } { \partial y }.
- Suppose z is an implicit function of x and y given by the equation.. yz + x ln(y) = z^2 Find \partial z / \partialx and \partial z / \partialy
- Use implicit differentiation to find partial z / partial x and partial z / partial y if ln (x y + y z + x z) = 5, (x greater than 0, y greater than 0, z greater than 0).
- 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
- Let cos (x z) + e^{x y z} = 2. Find partial z / partial x and partial z / partial y.
- 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.
- 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.
- Use implicit differentiation to find partial z / partial x. 8 x y^2 + 6 y z^2 + 10 z x^2 = 7.
- Find partial z / partial y if xyz = cos(2x + 3y -z).
- Given \ln(xy+yz+xz)=5, (x \gt 0, y \gt 0, z \gt 0) , use implicit differentiation to find \frac{\partial z }{\partial x} and \frac{\partial z }{\partial y}
- Given ln (x + y + z) - 2 y + z = 0. Find {partial z} / {partial y} at the point (1, 0, 0) by implicit differentiation.
- If z=(x+y)e^{x+y}, x=u, y= ln (v), find partial z/partial u and partial z/partial v.
- Given the equation x z^2 = e^{y z} where z is a function of x and y. Find partial z / partial x and partial z / partial y and the tangent plane at (1, 0, 1).
- Use implicit differentiation to find ( \partial z )/( \partial x ) where e^{z} = xz^{5} sin(y).
- 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 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}
- If z=xe^y, \ x=u^2+v^2, \ y=u^2-v^2, use the Chain Rule to find \frac{\partial z}{ \partial u} and \frac{\partial z}{ \partial v}.
- Suppose z depends on x and y via the equation cos(xyz) = x^2y^4 + xz^3 - pi^3 + 1 and x and y are independent variables. Using implicit differentiation to find partial z / partial x at (x, y, z) = (1, 0, pi).
- 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.
- Find partial f / partial x and partial f / partial y, if f(x, y) = x^2y - y^2 sin (4x).
- Use implicit differentiation to find the partial of z with respect to x and the partial of z with respect to y of x^2 + y^2 + z^2 = 3xyz
- Given z^3 - xy + yz + y^3 - 2 = 0. Find Partial Differential z/ Partial Differential x and Partial Differential z/ Partial Differential y using implicit differentiation. by treating z as an implicit f
- Use the Chain Rule to find \partial z/\partial s and \partial z/ \partial t . z = e^{-xy} \; \cos y, \;x = s/t, \;y = t/s .
- Find the first partial derivatives of the function. z = (3 x + 8 y)^5 partial z/partial x = partial z/partial y =
- 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 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.
- Find the partial derivative z = (e^{(2 x)}) * sin y a. partial z / partial x b. partial z / partial y
- If u = 4x^3 + 3y^3 + x^2y+ xy^2, find \partial u/ \partial x and \partial u/ \partial y.
- Let z = 4xy + e^{xy} where x = 2s + t^2, y = s^2 + 2t. \\ Find the partial derivatives of z in terms of the variables s and t.
- Given z = \frac{4x^2y^6 - y^6}{11xy - 5} , find the partial derivative z_y
- Assume z is a function of x and y. Find \partial z/ \partial x using the chain rule 3x^2 + 8z^2 + 9y^2 = 26.
- 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 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.
- 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 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.
- { Z = \sqrt{(x^2+y^2)} + xy } Find the partial derivatives: x, xx, y, yy, xy.
- 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.
- 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.
- Find the partial derivatives Z_x \ and \ Z_y \ for \ z =e^y \ sin(xy)
- If z = tan^(-1)(x/y) and x = u^2 + v^2 and y = u^2 - v^2, find the partial derivatives, partial z / partial u and partial z / partial v using the chain rule. (Express in terms of functions of u and v)
- Suppose that u(x, y) = - 5x^2 + 8xy + y^3 and v(x, y) = - 5x^3 + 3xy - 9y^2, partial f / partial u (4, -11) = - 5 and partial f / partial v (4, -11) = 5. Let g(x, y) = f(u(x, y), v(x, y)). Then find partial g / partial x (1, 1).
- 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)
- Consider f (x, y, z) = x / {y - z}. Compute the partial derivative below. {partial f} / {partial x}_{(2, -1, 3)}.
- Let x^{2} + 2y^{2} + 3z^{2} = 1. Solve for z and then use partial derivative to find \frac{\partial z}{\partial y}.
- If w=x^2+y^2, where x=r-s and y=r+s, then find partial w/partial r.
- Calculate the derivative using implicit differentiation: partial w/partial z, (x^4)w + w^4 + wz^2 + 3yz = 0. partial w/partial z =
- Find both first partial and the four second partial derivatives of the following functions: (a) z = 3xy^2 (b) z = x^2+3xy^3 (c) z = ln (x-y)
- 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 x 2 + 2 y 2 + 3 z 2 = 1 . a. Solve for z and then use partial derivative to find ? z ? y b. Find ? z ? y by implicit differentiation technique. c. Find ? z ? y by using Implicit Function
- Suppose z = x^2 sin y, x = 3 s^2 + t^2, y = 8 s t. A. Use the chain rule to find {partial z} / {partial s} and {partial z} / {partial t} as a function of x, y, s, and t. B. Find the numerical values of {partial z} / {partial s} and {partial z} / {partial
- 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 partial^2 z/partial x^2 for z=y^3-6xy^2-3.
- Find the second partial derivative z_{xy} of z = x^{2}y^{2} + 3xy + y^{3}
- 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.
- Find all the second partial derivatives of z = x^2 y +x sqrt y.
- Consider the following. Function: w = y^3- 2x^2y, \;x = e^s,\; y = e^t Point: s = 0, t = 3 Find \partial w /\partial s
- Calculate the derivative, partial w / partial z, using implicit differentiation: x^5 w + w^9 + w z^2 + 7 y z = 0.
- 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,
- Given z = x^4 + x^2y, x = s + 2t - u, y = stu^2 use the Chain Rule to find frac{partial z}{ partial s}, frac{partial z}{partial t}, and frac{partial z}{partial u} when s = 4, t = 2, and u
- Given z = 3x - x^2y^2, solve for \frac{\partial z}{\partial x} and \frac{\partial z}{\partial y} using partial derivatives.
- Calculate the derivative using implicit differentiation: partial w partial z, x^4w+w^7+wz^2+8yz=0
- Calculate the derivative using implicit differentiation: {partial w} / {partial z}, x^4 w + w^2 + w z^2 + 3 y z = 0
- 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).
- 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 /
- Calculate the derivative using implicit differentiation:{partial w / partial z}, {x^2w+w^8+wz^2+9yz=0}. Find dw/dz
- Suppose z = x^2 sin y, x = -3s^2 + 3 t^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
- 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)
- If f(x, y) = (x - y)/(x + y) then partial f/partial x is _____.
- 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).
- 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).
- 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
- 2-evaluate the indicated partial derivative z = (x2+5x-2y)8; dz/dx, dz/dy ?
- Using Chain Rule, to find (partial of z)/(partial of s), (partial of z)/(partial of t) and (partial of z)/(partial of u) if z = x^4 + x^2y, x = s + 2t u, y = stu^2; when s = 2, t = 1, u = 4.
- Given z = 2x^3 ln y + xy^2, find frac{partial^2z}{partial y partial x}.
- Consider the function f (x, y, z) = x cos (z) - e^{y z} near the point (1, 2, 1) First, find the value of the function at (1, 2, 1). Then, find the partial derivatives f_x, f_y, and f_z using the dif
- Find the second partial derivatives of Z = F(x,y) = 3x^{5y^3} - 3x + 4y + sin(xy).
- Suppose z = x^2 sin y, x = 2 s^2 + 1 t^2, y = -6 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. B. Find the numerical values of parti
- For f (x, y, z) = 2 x^2 - 3 x y^2 + 2 y^3, compute {partial f} / {partial x} (x, y).
- Find (partial w/partial y)_x at point (w, x, y, z) = (4, 3, 1, -1) if w = x^2 y^2 + yz - z^2 and x^2 + y^2 + z^2 = 6.
- 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
- Consider the equation xz^2+6yz-2ln(z)=3 as defining z implicitly as a function of x and y. The values of \partial z/ \partial x and \partial z/\partialy at (-3,1,1) are
- 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.