Use a chain rule to find the values of delta z / delta r_r = 12, theta = pi / 6 and delta z /...

Question:

Use a chain rule to find the values of {eq}\displaystyle \frac {\partial z}{\partial r}_{ r = 12,\ \theta = \frac {\pi}{6}} {/eq} and {eq}\displaystyle \frac {\partial z}{\partial \theta}_{ r = 12,\ \theta = \frac {\pi}{6}} {/eq}.

If {eq}z = xye^{\frac {x}{y}},\ x = r \cos \theta,\ \text{and}\ y = r \sin \theta {/eq}.

Chain Rule for Partial Derivatives:

Solving the problem above is an exercise in meticulous calculation. Not much to think about here, we just need to carefully compute the partials, then use the chain rule for partial derivatives:

{eq}\begin{align*} \frac{\partial z}{\partial s} &= \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \end{align*} {/eq}

Answer and Explanation: 1

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We need the partials of {eq}z {/eq} at the point for both partials. The partials of {eq}z {/eq} are

{eq}\begin{align*} \frac{\partial z}{\partial...

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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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