Preview Activity 2.4.1.
Consider the function \(f(x) = \tan(x)\text{,}\) and remember that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\text{.}\)
(a)
What is the domain of \(f\text{?}\)
(b)
Use the quotient rule to show that one expression for \(f'(x)\) is
\begin{equation*}
f'(x) = \frac{\cos(x)\cos(x) + \sin(x)\sin(x)}{\cos^2(x)}\text{.}
\end{equation*}
(c)
What is the Fundamental Trigonometric Identity? How can this identity be used to find a simpler form for \(f'(x)\text{?}\)
(d)
Recall that \(\sec(x) = \frac{1}{\cos(x)}\text{.}\) How can we express \(f'(x)\) in terms of the secant function?
(e)
For what values of \(x\) is \(f'(x)\) defined? How does this set compare to the domain of \(f\text{?}\)
