1. **State the problem:** Find the derivative of the function $f(x) = \tan^2 x$. 2. **Recall the formula:** The derivative of $\tan x$ is $\sec^2 x$. For a function $g(x)^2$, the derivative is $2 g(x) g'(x)$ by the chain rule. 3. **Apply the chain rule:** Let $g(x) = \tan x$, then $$f(x) = (g(x))^2$$ So, $$f'(x) = 2 g(x) g'(x) = 2 \tan x \cdot \sec^2 x$$ 4. **Simplify the expression:** $$f'(x) = 2 \tan x \sec^2 x$$ 5. **Interpretation:** The derivative of $\tan^2 x$ is $2 \tan x \sec^2 x$. This means the rate of change of $\tan^2 x$ at any point $x$ depends on both $\tan x$ and $\sec^2 x$ at that point. **Final answer:** $$\boxed{f'(x) = 2 \tan x \sec^2 x}$$