# 1.) Use the Chain Rule to find dz/dt. z = cos(x + 8y), x = 7t^5, y = 6/t. 2.) Use the Chain Rule...

## Question:

1.) Use the Chain Rule to find {eq} \partial z/ \partial t {/eq}.

{eq}z = cos(x + 8y) \quad x = 7t^5 \quad y = 6/t {/eq}

2.) Use the Chain Rule to find {eq} \partial z/ \partial s {/eq} and {eq} \partial z/ \partial t {/eq}.

{eq}z = x^9y^9 \quad x = s \cos t \quad y = s \sin t {/eq}

3.) Use the Chain Rule to find {eq} \partial z/ \partial s {/eq} and {eq} \partial z/ \partial t {/eq}.

{eq}z = \arcsin(x - y) \quad x = s^2 + t^2 \quad y = 1 - 4st{/eq}

## Chain Rule

To find the partial {eq}\frac{\partial z}{\partial t} {/eq} use the fact that {eq}x:=x(t,s) {/eq} and {eq}y:=y(t,s) {/eq} are functions in terms of t and s, then apply the identity

{eq}\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t} {/eq}

To compute a partial derivative {eq} \frac{\partial z}{\partial x} {/eq} hold y constant and taking the derivative with respect to x.

And, vise versa for computing the partial {eq} \frac{\partial z}{\partial y} {/eq}.

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Question 1

{eq}\begin{align*} \frac{\partial z}{\partial t} &= \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+ \frac{\partial... The Chain Rule for Partial Derivatives

from

Chapter 14 / Lesson 4
35K

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.