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


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

Become a member to unlock this answer! Create your account

View this answer

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...

See full answer below.

Learn more about this topic:

The Chain Rule for Partial Derivatives


Chapter 14 / Lesson 4

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.

Related to this Question

Explore our homework questions and answers library