Chapter 2.5, Problem 38ES

True-False Determine whether the statement is true or false. Explain your answer.

Suppose that $g\left(x\right)=f\left(x\right)\sec x$ , where $f\left(0\right)=8$ and $f^{\prime }\left(0\right)=-2$ . Then

$g^{\prime }\left(0\right)=\underset{h\to 0}{\lim }\frac{f\left(h\right)\sec h-f\left(0\right)}{h}=\underset{h\to 0}{\lim }\frac{8\left(\sec h-1\right)}{h}$

$=8\cdot {\left.\frac{d}{dx}\left[\sec x\right]\right|}_{x=0}=8\sec 0\tan 0=0$