# Find the derivative of the function. g(theta) = (cos(8 theta))^9

## Question:

Find the derivative of the function.

{eq}g(\theta) = (cos(8 \theta))^9 {/eq}

## Chain Rule:

For functions that are expressed as a composition of functions, we apply the derivative to the function by applying the chain rule. The formula for the chain rule is given by {eq}\displaystyle \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} {/eq}.

For the given function, we must apply the chain rule, {eq}\displaystyle \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} {/eq}, in order to correctly find the derivative of the function.

{eq}\begin{align} \displaystyle g(\theta) &= \cos^9(8\theta)\\ g'(\theta) &= \frac{d\ \cos^9(8\theta)}{d\ \cos(8\theta)} \frac{d\ \cos(8\theta)}{d\ (8\theta)}\frac{d(8\theta)}{d\theta}\\ &= 9\cos^8(8\theta) \cdot -\sin(8\theta) \cdot 8\\ &= -72\cos^8(8\theta)\sin(8\theta) \end{align} {/eq}

Using the Chain Rule to Differentiate Complex Functions

from Math 104: Calculus

Chapter 8 / Lesson 6
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