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}...
Question:
If {eq}\displaystyle z = x y^2 + x^2 y {/eq} where {eq}x = 3 u^2 - 2 v + 1 {/eq} and {eq}y = 2 u + v^3 {/eq}, find {eq}\displaystyle\dfrac {\partial z} {\partial u} {/eq} and {eq}\dfrac {\partial z} {\partial v} {/eq}.
Finding Partial Derivatives:
For finding the partial derivatives of the two combined functions, we can use the derivative rule called the chain rule. The chain rule for a single independent variable is {eq}\displaystyle \dfrac{\partial f(u)}{\partial x}=\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial x} {/eq}.
The chain rule for two independent variables is {eq}\displaystyle \dfrac{\partial z(x, y) }{\partial u} =\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u} {/eq}.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerThe given function is {eq}\displaystyle z = x y^2 + x^2 y {/eq}, where {eq}\displaystyle x = 3 u^2 - 2 v + 1 {/eq}, and {eq}\displaystyle y = 2 u...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
The Chain Rule for Partial Derivatives
from
Chapter 14 / Lesson 4
37K
This 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
- Use the equations to find partial z / partial x and partial z / partial y. x^2 + 8 y^2 + 7 z^2 = 1.
- 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
- If z = x^2y - y^2\sin^{-1} x, find partial z/partial y at (1, 2).
- Let z = {x y} / {4 y^2 - 4 x^2}. Then, find: (a) {partial z} / {partial x} (b) {partial z} / {partial y}
- Find partial f / partial x and partial f / partial y, if f(x, y) = x^2y - y^2 sin (4x).
- 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
- 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.
- Use implicit differentiation to find partial z/ partial x and partial z/partial y. x^2 + 2y^2 + 7z^2 = 8
- 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 e 2 z = x y z
- 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 partial^2 z/partial y partial x for z=y^3-6xy^2-3.
- Find partial^2 z/partial x partial y for z=y^3-6xy^2-3.
- 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 x for 3 x z^2 - e^4x cos 4z - 3 y^2 = 4. Do not simplify.
- Let g (x, y, z) = square root {x^2 + y^4 + z^3 - xyz}. Find the following: partial^2 f / partial x partial z and f_x (1, 2, 1).
- 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.
- Find partial z / partial y for xy^2 + ln(x^2y) + z squareroot x - squareroot 3 = 0 by performing the implicit differentiation.
- 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
- 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 x and partial z / partial y of the following function. z = (x^2 e^ {3 y} + y^2 e^ {3 x}).
- Find partial z over partial x and partial z over partial y by implicit differentiation if ye^x - 5 sin 3z = 3z.
- 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 / partial x + partial z / partial y, if sin (2 x y z) = x^3 + y^3 + z^3.
- Let f = f(x, y), x = s + t and y = s - t. Show that (partial f/partial x)^2 + (partial f/partial y)^2 = (partial f/partial s)(partial f/partial t).
- Let cos (x z) + e^{x y z} = 2. Find partial z / partial x and partial z / partial y.
- 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 _
- 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 }.
- 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).
- Find partial z / partial x and partial z / partial y, if x y z = tan (x + y+ z).
- 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
- 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.
- 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} and {partial z} / {partial y} if e^{-x y} - 3 z + e^z = 0.
- Use the three-dimensional implicit equation below to answer the following: x 2 y 3 z 4 = 5 x + 40 y + 35 z 1. Find partial z partial x 2. Find partial z partial y
- Find {partial z}/{ partial x}, { partial z}/{ partial t} where z = x^y + x, x = sqrt{s + t} ; and ; y = s^2t
- 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
- If z=(x+y)e^{x+y}, x=u, y= ln (v), find partial z/partial u and partial z/partial v.
- 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.
- Let z = ^{u - 2v}, and u = xy^2, v = x^{2}y, find \frac{\partial z }{\partial x} and \frac{\partial z }{\partial y}.
- 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).
- If u = 4x^3 + 3y^3 + x^2y+ xy^2, find \partial u/ \partial x and \partial u/ \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}.
- 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).
- 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.
- If w=x^2+y^2, where x=r-s and y=r+s, then find partial w/partial r.
- 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 the partial elasticities of z with respect to x, where z = ln(x^2 + y^2)
- Find partial w / partial s and partial w / partial t for w = x^2 + y^2 - 2 z^2 + 2 x y + 3 x z - 4 y z, where x = 4 s + t, y = 3 s - t , and z = -2 s + 5 t. Write your answers as functions only of s a
- 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)
- Find the first partial derivatives of the function. z = (3 x + 8 y)^5 partial z/partial x = partial z/partial y =
- Use implicit differentiation to find ( \partial z )/( \partial x ) where e^{z} = xz^{5} sin(y).
- 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
- Given ln (x + y + z) - 2 y + z = 0. Find {partial z} / {partial y} at the point (1, 0, 0) by implicit differentiation.
- 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 z / partial y if xyz = cos(2x + 3y -z).
- 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.
- 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
- 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.
- 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).
- Find the partial derivative z = (e^{(2 x)}) * sin y a. partial z / partial x b. partial z / partial y
- 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
- 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.
- 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}
- Let r = sin (p / q), p = square root {x y^2 z^3}, q = square root {x + 2 y + 3 z}. Use the chain rule to find partial r / partial x, partial r / partial y.
- 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
- 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}
- 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 .
- Let x^{2} + 2y^{2} + 3z^{2} = 1. Solve for z and then use partial derivative to find \frac{\partial z}{\partial y}.
- 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 (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).
- 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
- Find partial^2 z/partial x^2 for z=y^3-6xy^2-3.
- 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).
- 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 : 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 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.
- 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 (\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)}.
- 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,
- Let f(x,y) = xe^{x^3y}. Determine \frac{\partial^2 f}{\partial y^2} and \frac{\partial^2 f}{\partial x \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.
- Calculate the derivative using implicit differentiation: partial w/partial z, (x^4)w + w^4 + wz^2 + 3yz = 0. partial w/partial z =
- 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).
- 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 = 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 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 (\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 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.
- For f (x, y, z) = 2 x^2 - 3 x y^2 + 2 y^3, compute {partial f} / {partial x} (x, y).
- 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
- Determine dy/dx of x^2y^2-y+cos x=0 using dy/dx= partial f/partial x/partial f/partial y.
- Find partial f / partial x, partial f / partial y, and partial f / partial z. f (x, y, z) = 21 e^ x y z
- 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.
- Given z = 2x^3 ln y + xy^2, find frac{partial^2z}{partial y partial x}.
- 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
Explore our homework questions and answers library
Browse
by subject