Question

(1 point)
let
f(x)=\frac{6 - 7\tan(x)}{sec(x)}.
find:
f(x)=
f(4pi)=

Explanation:

Step1: Simplify the function

Use the identities \(\tan(x)=\frac{\sin(x)}{\cos(x)}\) and \(\sec(x)=\frac{1}{\cos(x)}\).

$$
LATEXBLOCK0
$$

Step2: Differentiate the function

Differentiate \(y = 6\cos(x)-7\sin(x)\) using the rules \((\cos(x))^\prime=-\sin(x)\) and \((\sin(x))^\prime=\cos(x)\).

$$ f^\prime(x)=-6\sin(x)-7\cos(x) $$

Step3: Evaluate \(f^\prime(4\pi)\)

Substitute \(x = 4\pi\) into \(f^\prime(x)\). Since \(\sin(4\pi)=0\) and \(\cos(4\pi)=1\).

$$ f^\prime(4\pi)=-6\times0 - 7\times1=-7 $$

Answer:

\(f^\prime(x)=-6\sin(x)-7\cos(x)\)
\(f^\prime(4\pi)=-7\)