7. - / 1 points find the derivative of the function. g(\\theta)=7\\cos^{4}(\\theta) g(\\theta)=

Explanation:

Step1: Apply chain - rule

Let $u = \cos(\theta)$, then $g(\theta)=7u^{4}$. The chain - rule states that $\frac{dg}{d\theta}=\frac{dg}{du}\cdot\frac{du}{d\theta}$. First, find $\frac{dg}{du}$. $\frac{dg}{du}=\frac{d}{du}(7u^{4}) = 28u^{3}$

Step2: Find $\frac{du}{d\theta}$

Since $u = \cos(\theta)$, then $\frac{du}{d\theta}=-\sin(\theta)$

Step3: Combine using chain - rule

$\frac{dg}{d\theta}=\frac{dg}{du}\cdot\frac{du}{d\theta}=28u^{3}\cdot(-\sin(\theta))$. Substitute $u = \cos(\theta)$ back in. $g'(\theta)=28\cos^{3}(\theta)\cdot(-\sin(\theta))=- 28\cos^{3}(\theta)\sin(\theta)$

Answer:

$-28\cos^{3}(\theta)\sin(\theta)$