Use the chain rule to find expressions for partial z by partial u and partial z by partial w if z...
Question:
Use the chain rule to find expressions for {eq}\frac{\partial z}{\partial u}{/eq} and {eq}\frac{\partial z}{\partial w}{/eq} if
{eq}z=f(x,y), \quad x=h(u,w), \quad y=g(w,t){/eq}
(Just in case, there are no typos in the above formulas. In particular, {eq}x{/eq} does not depend on {eq}t{/eq} and {eq}y{/eq} does not depend on {eq}u.{/eq})
Chain Rule:
The chain rule is a mathematical operation that is used in differential calculus to differentiate composite functions. Chain rule to be applicable, the composition of the composite function should be differentiable.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerGiven data
- The given function are;
{eq}z = f\left( {x,y} \right)............\left( 1 \right) {/eq}
{eq}x = h\left( {u,w} \right)............\left(...
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 Chain Rule to find \partial z/\partial s and \partial z/ \partial t . z = e^{-xy} \; \cos y, \;x = s/t, \;y = t/s .
- Use chain rule to find the indicated partial derivative. z = x^2 + x y^3, x = uv^2 + v^3, y = v e^u. Find {partial z} / {partial u} when u = 2, v = 1.
- Use Chain Rule to find expression for partial z / partial u and partial z / partial w, if z = f (x, y), x = (u, w), g = (w, t).
- 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
- 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
- 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}.
- Use the Chain Rule to find the indicated partial derivatives: z=x^4+x^2y, x=s+2t-u, y=stu^2; the partial derivative of z with respect to x, the partial derivative of z with respect to t, and the par
- Use the chain rule to find partial z / partial s and partial z / partial t, where z=x^2+xy+y^2, x=3s + 4t, y=5s + 6t.
- 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 z = x^2\sin y, x = -2s^2+0t^2, y = -6st. (a) Use the chain rule to find the partial derivative of z with respect to s and the partial derivative of z with respect to t as functions of x, y, s,
- Use the Chain Rule to find partial z / partial s and partial z / partial t. z = x^3 y^9, x = s cos t, y = s sin t.
- Use the Chain Rule to find partial z / partial s and partial z / partial t. z = (x - y)^7, x = (s^2)t, y = s(t^2).
- Use the chain rule to find partial z/partial s and partial z/partial t. z=tan (u/v), u=6s+5t, v=5s-6t.
- Use the chain rule to find partial f / partial u, and partial f / partial v, if f(x, y) = cos(xy), x(u, v) = u + v and y(u, v) = u - v.
- Use the Chain Rule to find {partial z/partial s} and {partial z/partial t}. z = tan (u/v), u = 4 s + 7 t, v = 7 s - 4 t
- 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.
- Use the chain rule to find the partial derivative of z with respect to x given that z = u^2 \ cos \ 4v; u = x^2y^3; \ and \ y = x^3 + y^3
- Use the chain rule to find the partial derivative of z with respect to y given that z = u^2 \ cos \ 4v; u = x^2y^3; and \ v = x^3 +y^3
- Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2 y, x = s + 2t - u, y = stu^2; partial z/partial s, partial z/partial t, partial z/partial u when s = 3, t = 5, u = 2.
- Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2 y, x = s + 2t - u, y = stu^2; partial z/partial s, partial z/partial t, partial z/partial u when s = 3, t = 2, u = 1.
- Use the Chain Rule to find the indicated partial derivatives. z = x^2 + x y^3, x = u v^3 + w^2, y = u + v e^w {partial z / partial u}, {partial z / partial v}, {partial z / partial w} when u = 2, v
- Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2 y, x = s + 2 t - u, y = s t u^{22}; find partial z / partial s, partial z / partial t, partial z / partial u when s = 5, t =
- Use the Chain Rule to find the indicated partial derivatives. N = (p + q) / (p + r), p = u + vw, q = v + uw, r = w + uv; partial N / partial u, partial N / partial v, partial N / partial w when u = 7,
- Use the Chain Rule to find partial z/partial s and partial z/partial t. z = tan(u/v), u = 2s + 8t, v = 8s - 2t.
- Use the Chain Rule to find partial z / partial s and partial z / partial t, where: z = sin (-3 x) cos (8 y), x = t^2 - 3 s^2, y = 3 - s t.
- If u = x y + y z + z x ; x = r 2 + s ; y = r s ; z = r + s ; find the partial derivatives u r a n d u s by using the Chain-Rule.
- Use the Chain Rule to find the indicated partial derivatives. R = ln (u^2 + v^2 + w^2), u = x + 8y, v = 4x - y, w = 2xy; Compute partial R / partial x , partial R / partial y, when x = y = 4.
- Use the 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.
- Use the Chain Rule to find the indicated partial derivatives. z = x^3 + xy^4, x = uv^4 + w^3, y = u + ve^w. Find [{partial z} / {partial u}], [{partial z} / {partial v}], [{partial z} / {partial w}],
- Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2y, x = s + 2t - u, y = stu^2 when s = 2, t = 4, u = 5. Find: partial z / partial s , partial z / partial t , partial z / part
- Use the Chain Rule to find \partial z/\partial s and \partial z/\partial t. z=(x-y)^{9}, x=s^{2}t, y=st^{2}
- Use the Chain Rule to find \partial z / \partial s and \partial z/\partial t. z = x^7y^9,\;\;\;x = s\:cos(t),\;\;\;y = s\:sin(t) \frac{\partial z}{\partial s = _____ \frac{\partial z}{\partial t} = _____
- Use the Chain Rule to find the indicated partial derivatives. R = ln (u^2 + v^2 + w^2), u = x + 3 y, v = 8 x - y, w = 2 x y; partial R / partial x , partial R / partial y when x = y = 1.
- Find \frac{ \partial w }{ \partial s } and \frac{ \partial w }{ \partial t } by using the appropriate Chain Rule. w = xyz, x = s + 7t, y = s - 7t, z = st^2
- Use the Chain Rule to find the indicated partial derivatives. N = {p + q} / {p + r}, p = u + v w, q = v + u w, r = w + u v; {partial N} / {partial u}, {partial N} / {partial v}, {partial N} / {partial w} when u = 2, v = 3, w = 4
- Use the Chain Rule to find the indicated partial derivatives. u = x4 + yz, x = pr cos(theta), y = pr sin(theta), z = p + r , p = 1, r = 2, theta = 0 Find partial u/partial p , partial u/partial r , p
- Use the Chain Rule to evaluate the partial derivative at the point specified: partial f / partial u and partial f / partial v at (u, v) = (-1, -1), where f(x, y, z) = x^3 + yz^2, x = u^2 + v, y = u + v^2, z = uv.
- Use the Chain Rule to find (partial of z)/(partial of s) and (partial of z)/(partial of t) if z = tan(u/v), u = 7s + 2t, v = 2s - 7t
- Using Chain Rule, find (partial of z)/(partial of s) and (partial of z)/(partial of t) if z = tan(u/v), u = 7s + 2t, v = 2s 7t.
- If f (x, y) = (2 x - 5 y)^3, where x = s t^3 and y = s^2 t; use the chain rule to find partial f / partial s.
- Use the chain rule to find the indicated partial derivatives. z=x^4+x^2y, x=s+2t-u, y=stu^2; partial z/partial s, partial z/partial t, partial z/partial u when s=4, t=5, u=3.
- Use the chain rule to find partial z/partial s and partial z/partial t, where z = e^(xy) tan y, x = 5s + 1t, y = (4s)/(3t).
- Use the Chain Rule to find the indicated partial derivatives. z = x^4 + x^2y; x = s + 2t - u; y = stu^2 \partial z/\partial s, \partial z / \partial t, \partial z/\partial u when s = 5, t = 4 and u = 1.
- Use the Chain Rule to find the indicated partial derivatives. R = ln (u^2 + v^2 + w^2) ; u = x + 8 y, v = 7 x - y, w = 3 x y ; {partial R / partial x}, {partial R / partial y} when x = y = 1. {partia
- Use the Chain Rule to find the indicated partial derivatives. R = ln (u^2 + v^2 + w^2), u = x + 9 y, v = 5 x - y, w = 2 x y; partial R / partial x, partial R / partial y when x = y = 1
- Use the Chain Rule to find the indicated partial derivatives. z = x^3 + xy^2, x = uv^2 + w^3, y = u + ve^w (partial z)/(partial u), (partial z)/(partial v), (partial z)/(partial w)when u = 2, v = 1, w
- Given. f(x, y, z) = xy + z^4 ; x = r + s - 4t ; y = 3rt ; z = s^11 Use the Chain Rule to calculate the partial derivatives partial f/ partial r, partial f/partial t.
- If w = x y^2 + y z - 3, x = r^2 s - t, y = r + 2 s t, z = r s - 2 t then by chain rule find {partial w} / {partial t}.
- Use the Chain Rule to find \partial z/\partial s and \partial z/\partial t. Give your answer in terms of s and t. z=\arcsin(x-y), x=s^2+t^2, y=4-3st..
- Use the Chain Rule to evaluate the partial derivative at the point specified. partial f / partial u and partial f / partial v at (u, v) = (-1, 1), where f (x, y, z) = x^3 + y z^2, x = u^2 + v, y = u + v^2, z = u v.
- Use the Chain Rule to evaluate the partial derivative at the point specified. partial g / partial s at s = 2, where g(x, y) = x^2 - y^2, x = s^2 + 3, y = 3 - 2 s. partial g / partial s|_{s=2} =
- Calculate the derivative using implicit differentiation: partial w/partial z, (x^4)w + w^4 + wz^2 + 3yz = 0. partial w/partial z =
- Use the Chain Rule to find the partial derivative of z with respect to r, that is \frac{\partial z}{\partial r}, and the partial derivative of z with respect to \theta, that is \frac{\partial z}{\partial \theta}, given the surface z=x^5+x^2y+y^3x where x
- 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
- Find \frac{ \partial z}{ \partial \tau} \ and \ \frac{ \partial z}{ \partial s} }] using the chain rule . z=x^2y^2,x=s \tau,y=2s+ \tau
- 1. (a) Use chain rule to find the partial derivatives partial z / partial u and partial z / partial v of z = e^{x^2 y} where x (u, v) = square root of {uv} and y(u, v) = 1 / v. (b) Express partial z / partial u and partial z / partial v only in terms of u
- Let f (x, y, z) = x^2 + 4 y^2 + z Use Chain Rule to find partial f / partial u when (u, v) = (0, 1), where x = u v, y = 2 v + 3 and z = u + v.
- Use the Chain Rule to find partial z / partial s and partial z / partial t. z = e^{x + 4y}, x = s / t, y = t / s.
- Use the chain rule to find the indicated partial derivatives. u = square root {r^2 + s^2}, r = y + x cos (t), s = x + y sin (t): partial u / partial x, partial u / partial y, partial u / partial t, wh
- Use the Chain Rule to find the indicated partial derivatives. M= xe^(y- z^3), x = 8uv, y = u - v, z = u + v; find (partial M)/(partial u), (partial M)/(partial v) when u = 2, v = -2
- Use the Chain Rule to find the indicated partial derivatives. z = x^3 + xy^2, x = uv^2 + w^3, y = u + v*e^w (partial z)/(partial u), (partial z)/(partial v), (partial z)/(partial w) when u = 2, v = 1, w = 0
- 2-evaluate the indicated partial derivative z = (x2+5x-2y)8; dz/dx, dz/dy ?
- Consider the function f(x,y) = 4xy/e^{x^2} +e^{y^2}: Find partial f/partial x and partial f/partial y using the chain rule.
- (a) Use chain rule to find the partial derivatives {partial z} / {partial u} and {partial z} / {partial v} of z = e^{x^2 y}, where x(u, v) = square root {u v} and y (u, v) = 1 / v. (b) Express {partial z} / {partial u} and {partial z} / {partial v} only i
- Write a chain rule formula for the following derivative. {partial u} / {partial t} for u = f (v) ; v = h (s, t). (a) {partial u} / {partial t} = {partial v} / {partial t} (b) {partial u} / {partial t}
- Given z = \frac{4x^2y^6 - y^6}{11xy - 5} , find the partial derivative z_y
- Use the Chain Rule to evaluate the partial derivative at the point specified. partial f / partial s at (r, s) = (1, 0), where f (x, y) = ln (x y), x = 3 r + 2 s, y = 5 r + 3 s.
- Use the Chain Rule to evaluate the partial derivative at the point specified: partial f / partial s at (r, s) = (1, 0), where f(x, y) = ln(xy), x = 3r + 2s, y = 5r + 3s.
- Use the Chain Rule to evaluate the partial derivative at the point specified. \frac{\partial f }{\partial u} and \frac{\partial f }{\partial v} at (u, v) = (- 1, - 1), where f(x, y, z) = 5x^3 + 8yz^2
- Use the Chain Rule to evaluate the partial derivative at the point specified. {partial g} / {partial s} at s = 4, where g (x, y) = x^2 - y^2, x = s^2 + 1, y = 1 - 2 s.
- Find partial z / partial y for xy^2 + ln(x^2y) + z squareroot x - squareroot 3 = 0 by performing the implicit differentiation.
- Use the chain Rule to find the indicated partial derivatives. z=x^4+x^2y, x=s+2t-u, y=stu^2; fraction {partial z}{partial s},fraction {partial z}{partial t},fraction {partial z}{partial u}
- Find the first partial derivatives of the function. z = (3 x + 8 y)^5 partial z/partial x = partial z/partial y =
- Use the Chain Rule to find the following partial derivatives when u=1, v=1, and w=0 given z=x^3 +x(y^4), x=u(v^4) + w^3, and y=u+v(e^w) a) the partial of z with respect to u b) the partial of z wi
- Calculate the derivative, partial w / partial z, using implicit differentiation: x^5 w + w^9 + w z^2 + 7 y z = 0.
- Find partial z / partial v using the chain rule if z = sin (x / y), x = ln(u), and y = v. Put the answer in terms of u and v.
- Use the chain rule to find partial z/partial x and partial z/partial t, where z=e^{xy} tan (y), x=s+4t, y=4s/4t.
- Use implicit differentiation to find ( \partial z )/( \partial x ) where e^{z} = xz^{5} sin(y).
- Use the Chain Rule to find (partial of z)/(partial of s) and (partial of z)/(partial of t) if z = x^7y^7, x = s cos t, y = s sin t.
- Using Chain Rule, find (partial of z)/(partial of s) and (partial of z)/(partial of t) if z = x^7y^7, x = s cos t, y = s sin t.
- For the functions w = xy + y^2 + xz, x = 2u + v, y = 2u - v, and z = uv, express partial w / partial u and partial w / partial v using the chain rule and by expressing w directly in terms of u and v before differentiating.
- 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 . Please use * for multiplication between all factors.) z = x^8y^
- Let \ z = ue^{(u^2-v^2)} , where \ u = 2x^2 + 3y^2 \ and \ v = 3x^2 - 2y^2 . Use \ the \ chain \ rule \ to \ find\ \frac {\partial f}{\partial x}, and\ \frac {\partial f}{\partial y}.
- Use the chain rule to find partial z/partial s and partial z/partial t, where z = e(xy)tan y, x = 5s+2t, y = 6s/4t
- Use the chain rule to find partial z/partial s and partial z/partial t, where z = x^2 + xy + y^2, x = 5s + 8t, y = 6s + 2t.
- 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
- Use the Chain Rule to calculate the partial derivatives partial f / partial t and partial f / partial s. Express the answer in terms of the independent variables. (a) f (x, y) = cos(x - y), x = 3 t -
- Let x^{2} + 2y^{2} + 3z^{2} = 1. Solve for z and then use partial derivative to find \frac{\partial z}{\partial y}.
- Use the chain rule to find the indicated partial derivatives. P = \sqrt {u^2 + v^2 + w^2}, u = xe^y, v = ye^x, w = e^{xy}; \frac {\partial P}{\partial x}, \frac {\partial P}{\partial y}, when x = 0, y = 2
- Use the Chain Rule to find the indicated partial derivatives. P = sqrt{u^2 + v^2 + w^2}, u = xe^y, v = ye^x, w = e^xy; {partial P}/{partial x}, {partial P}/{partial y} when x = 0, y = 5
- Use the Chain Rule to find the indicated partial derivatives. u = sqrt(r^2+s^2), r = y + x cos(t), s = x + y sin(t);(partial u)/(partial x), (partial u)/(partial y), (partial u)/(partial t) when x = 1
- 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.
- Use the Chain Rule to find the indicated partial derivatives.\frac{\partial z}{\partial s},\frac{\partial z}{\partial t},\frac{\partial z}{\partial u} z=x^{4}+x^{2}y, x=s+2t-u, y=stu^{2}; when s=4, t=
- Use the chain rule to find \curly z/ \curly s and \curly z/ \curly t, where z = e^xy tany, x = 1s + 3t, y = 5s/5t First the pieces: And putting it all together:
- Use the Chain Rule to calculate the partial derivatives \displaystyle \frac { \partial g } { \partial u } and \displaystyle \frac { \partial g } { \partial v } where g ( x , y ) = \operatorname {
- Use the Chain Rule to find the indicated partial derivatives. P = square root {u^2 + v^2 + w^2}, u = x e^y, v = y e^x, w = e^{x y}; partial P / partial x, partial P / partial y when x = 0, y = 5.
- Use the Chain Rule to find the indicated partial derivatives. P = square root {u^2 + v^2 + w^2}, u = x e^y, v = y e^x, w = e^{x y}; {partial P} / {partial x}, {partial P} / {partial y} when x = 0, y = 2
- Use the chain rule to find \frac{\partial z }{\partial s} and \frac{\partial z }{\partial t} when z = x^2+xy + y^2 , x=s+t, y =st .
- Use the Chain Rule to find \dfrac{\partial z}{\partial s} and \dfrac{\partial z }{\partial t}. \\ z = \tan\left(\dfrac{u}{v}\right),\ u = 8s _ 4t,\ u= 4s - 8t
Explore our homework questions and answers library
Browse
by subject