Question

find ( gleft(\frac{pi}{4}
ight) ) for ( g(\theta)=cos (3 \theta+pi) ).

Explanation:

Step1: Differentiate \(g(\theta)\) using chain rule

Let \(u = 3\theta+\pi\), then \(g(\theta)=\cos(u)\).
The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), and the derivative of \(u = 3\theta+\pi\) with respect to \(\theta\) is \(3\).
By the chain rule \(\frac{dg}{d\theta}=\frac{dg}{du}\cdot\frac{du}{d\theta}\), so \(g'(\theta)=- 3\sin(3\theta+\pi)\).

Step2: Substitute \(\theta=\frac{\pi}{4}\) into \(g'(\theta)\)

\(g'(\frac{\pi}{4})=-3\sin(3\times\frac{\pi}{4}+\pi)\).
Simplify the argument of the sine function: \(3\times\frac{\pi}{4}+\pi=\frac{3\pi}{4}+\pi=\frac{3\pi + 4\pi}{4}=\frac{7\pi}{4}\).
Since \(\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}\), then \(g'(\frac{\pi}{4})=-3\times(-\frac{\sqrt{2}}{2})\).

Answer:

\(\frac{3\sqrt{2}}{2}\)