Suppose that f is a differentiable function with f_x (8,0) = 8 and f_y (8, 0) = 7. Let w(u, v) =...

Question:

Suppose that {eq}f {/eq} is a differentiable function with {eq}f_x (8,0) = 8 {/eq} and {eq}f_y (8, 0) = 7 {/eq}.

Let {eq}\displaystyle w(u, v) = f (x(u,\ v),\ y(u,\ v)) {/eq} where {eq}x = 8 \cos u + 2 \sin v {/eq} and {eq}y = 8 \cos u \sin v {/eq}. Find {eq}w_v (0,\ 0) {/eq}.

Chain Rule for Derivatives:

One of the most important facet in mathematics is the way how mathematicians generalize common concepts so as to apply known findings to more complex problems. In this exxample, we shall illustrate how to use the generalization of chain rule to evaluate a partial derivative of a given function. In particular, for a function {eq}f(x(t),y(t)) {/eq}

one can solve the derivative of f with respect to t as

{eq}\partial_tf(x(t),y(t)) = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial t}. {/eq}

Answer and Explanation: 1

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For the function {eq}\displaystyle w(u, v) = f (x(u,\ v),\ y(u,\ v)) {/eq} where {eq}x = 8 \cos u + 2 \sin v {/eq} and {eq}y = 8 \cos u \sin...

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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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