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}...


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

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The 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...

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The Chain Rule for Partial Derivatives


Chapter 14 / Lesson 4

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

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