Let : z^4 = 8xe^{y/z} calculate the partial derivatives partial z / partial x , partial x / ...
Question:
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}
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View this answer{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
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Chapter 14 / Lesson 4
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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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