if ( f(x)=cos x - 2\tan x ), then ( f(x)=) ( f(5)=)

Explanation:

Step1: Differentiate (f(x))

Use the derivative rules: ((\cos x)'=-\sin x) and ((\tan x)'=\sec^{2}x). For (f(x)=\cos x - 2\tan x), by the sum - difference rule ((u - v)'=u' - v') (where (u = \cos x) and (v = 2\tan x)), and the constant multiple rule ((cf(x))'=cf'(x)) ((c = 2)). (f'(x)=(\cos x)'-2(\tan x)'=-\sin x-2\sec^{2}x).

Step2: Evaluate (f'(x)) at (x = 5)

Substitute (x = 5) into (f'(x)). (f'(5)=-\sin(5)-2\sec^{2}(5)). Since (\sec x=\frac{1}{\cos x}), then (f'(5)=-\sin(5)-\frac{2}{\cos^{2}(5)}). Using a calculator (in radian mode): (\sin(5)\approx - 0.9589), (\cos(5)\approx0.2837), (\cos^{2}(5)\approx0.0805). (f'(5)=-(- 0.9589)-\frac{2}{0.0805}=0.9589 - 24.8447=-23.8858).

Answer:

(f'(x)=-\sin x - 2\sec^{2}x); (f'(5)\approx - 23.89)