Derivative of a product of trig functions

Apply the product rule: $f'(x) = \frac{d}{dx}[\tan x]\cdot \sec x + \tan x \cdot \frac{d}{dx}[\sec x]$. Substituting the known derivatives $\frac{d}{dx}[\tan x] = \sec^2 x$ and $\frac{d}{dx}[\sec x] = \sec x \tan x$ gives $f'(x) = \sec^2 x \cdot \sec x + \tan x \cdot \sec x \tan x$.

Simplify each term: the first term is $\sec^3 x$ and the second term is $\sec x \tan^2 x$. Factoring out the common factor $\sec x$ gives $f'(x) = \sec x(\sec^2 x + \tan^2 x)$, which can also be left as $f'(x) = \sec^3 x + \sec x \tan^2 x$.