let\n$f(x)=-5\\cos x + 5\\tan x$\n$f(x)=$\n$f(\\frac{2\\pi}{3})=$\nquestion help: video message…

Explanation:

Step1: Differentiate each term

Use derivative rules: ((\cos x)'=-\sin x), ((\tan x)'=\sec^{2}x). (f'(x)=(-5\cos x)'+(5\tan x)'=5\sin x + 5\sec^{2}x)

Step2: Substitute (x = \frac{2\pi}{3})

First, find (\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}), (\sec\frac{2\pi}{3}=\frac{1}{\cos\frac{2\pi}{3}}=\frac{1}{-\frac{1}{2}}=- 2). (f'(\frac{2\pi}{3})=5\times\frac{\sqrt{3}}{2}+5\times(-2)^{2}=\frac{5\sqrt{3}}{2}+20)

Answer:

(f'(x)=5\sin x + 5\sec^{2}x); (f'(\frac{2\pi}{3})=\frac{5\sqrt{3}}{2}+20)