Question

let
$f(x)=-5\cos x + 5\tan x$
$f(x)=$
$f(\frac{2\pi}{3})=$
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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:

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