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

Explanation:

Step1: Simplify the function

Use the identities (\tan(x)=\frac{\sin(x)}{\cos(x)}) and (\sec(x)=\frac{1}{\cos(x)}). [ \begin{align*} f(x)&=\frac{6 - 7\tan(x)}{\sec(x)}\ &=(6 - 7\frac{\sin(x)}{\cos(x)})\cos(x)\ &=6\cos(x)-7\sin(x) \end{align*} ]

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)