Question

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:

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