Question

find the derivative of the function.
g(\theta)=5\cos^{4}(\theta)
g(\theta)= square

Explanation:

Step1: Identify outer - inner functions

Let $u = \cos(\theta)$, then $y = 5u^{4}$.

Step2: Differentiate outer function

Using the power rule $\frac{d}{du}(au^{n})=nau^{n - 1}$, for $y = 5u^{4}$, $\frac{dy}{du}=20u^{3}$.

Step3: Differentiate inner function

$\frac{du}{d\theta}=-\sin(\theta)$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{d\theta}=\frac{dy}{du}\cdot\frac{du}{d\theta}$. Substitute $u = \cos(\theta)$ back in. So $\frac{dy}{d\theta}=20\cos^{3}(\theta)\cdot(-\sin(\theta))$.

Answer:

$- 20\cos^{3}(\theta)\sin(\theta)$