Let W(s,t)=F(u(s,t),v(s,t)) where...

Question:

Let {eq}W\left( s,t \right)=F\left( u\left( s,t \right),v\left( s,t \right) \right) {/eq} where

{eq}\begin{align} & u\left( 1,0 \right)=-1,{{u}_{s}}\left( 1,0 \right)=9,{{u}_{t}}\left( 1,0 \right)=3 \\ & v\left( 1,0 \right)=-2,{{v}_{s}}\left( 1,0 \right)=4,{{v}_{t}}\left( 1,0 \right)=1 \\ & {{F}_{u}}\left( -1,-2 \right)=4,{{F}_{v}}\left( -1,-2 \right)=1 \\ \end{align} {/eq}

{eq}\begin{align} & {{W}_{s}}\left( 1,0 \right)=\left( ............ \right) \\ & {{W}_{t}}\left( 1,0 \right)=\left( ............ \right) \\ \end{align} {/eq}

Chain Rule:

We have to find the value of {eq}W_s\left( 1,0 \right) \text {and} W_t\left( 1,0 \right) {/eq} with the help of the given values. Function w has two variables u and v. The functions u and v has two variables s and t. So, the given composite function is solved by chain rule of differentiation, then substitute and simplify it to get the desired results.

Answer and Explanation: 1

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First, we will partially differentiate the composite function w with respect to s and we have $$\begin{align*} W\left( {s,t} \right) &= F\left(...

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The Chain Rule for Partial Derivatives

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Chapter 14 / Lesson 4
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When evaluating the derivative of composite functions of several variables, the chain rule for partial derivatives is often used. In this lesson, we use examples to explore this method.


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