Question

if $f(x)=\frac{\tan x - 4}{sec x}$
$f(x)=$
$f(4)=$
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Explanation:

Step1: Simplify the function

We know that \(\tan x=\frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). So \(f(x)=\frac{\tan x - 4}{\sec x}=\sin x-4\cos x\)

Step2: Differentiate the function

Using the derivative rules \((\sin x)'=\cos x\) and \((\cos x)'=-\sin x\).
For \(y = f(x)=\sin x-4\cos x\), by the sum - rule of differentiation \(y'=f'(x)=(\sin x)'-4(\cos x)'\)
\(f'(x)=\cos x + 4\sin x\)

Step3: Evaluate the derivative at \(x = 4\)

Substitute \(x = 4\) into \(f'(x)\):
\(f'(4)=\cos(4)+4\sin(4)\)

Answer:

\(f'(x)=\cos x + 4\sin x\)
\(f'(4)=\cos(4)+4\sin(4)\)