Let : z^4 = 8xe^{y/z} calculate the partial derivatives partial z / partial x , partial x / ...


Let : {eq}\displaystyle z^4 = 8xe^{y/z} {/eq} calculate the partial derivatives {eq}\displaystyle \frac {\partial z} { \partial x} ,\ \frac {\partial x} {\partial z} {/eq} using implicit differentiation ?

(a) {eq}\displaystyle \frac {\partial z} {\partial x} = \ ? {/eq}

(b) {eq}\displaystyle \frac {\partial x} {\partial z} = \ ? {/eq}

Chain Rule for Partial Derivatives:

From the mathematical Equation {eq}z=z(x,y), {/eq} where {eq}x=x(r,s) {/eq} and {eq}y=y(r,s) {/eq} then {eq}z=z(r,s) {/eq} and Chain Rule states that {eq}\displaystyle \dfrac{\partial z}{\partial r}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial r}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial r}\\ \displaystyle \dfrac{\partial z}{\partial s}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial s}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial s}.\\ {/eq}

Answer and Explanation: 1

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{eq}\displaystyle z^4 = 8xe^{y/z} {/eq}

Considering z=z(x, y).

Differentiate partially with respect to x keeping y as constant using chain rule,


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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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