Second Derivative Tan Square 52C92F

1. **State the problem:** Find the second derivative of the function $$f(x) = \tan^2 x$$. 2. **Recall the formula:** The first derivative of $$\tan x$$ is $$\sec^2 x$$. Using the chain rule, the derivative of $$\tan^2 x$$ is $$2 \tan x \cdot \sec^2 x$$. 3. **Find the first derivative:** $$ f'(x) = 2 \tan x \cdot \sec^2 x $$ 4. **Find the second derivative:** Differentiate $$f'(x)$$ using the product rule: $$ f''(x) = 2 \left( \frac{d}{dx} \tan x \cdot \sec^2 x + \tan x \cdot \frac{d}{dx} \sec^2 x \right) $$ 5. **Calculate each derivative:** - $$\frac{d}{dx} \tan x = \sec^2 x$$ - $$\frac{d}{dx} \sec^2 x = 2 \sec^2 x \tan x$$ (using chain rule) 6. **Substitute back:** $$ f''(x) = 2 \left( \sec^2 x \cdot \sec^2 x + \tan x \cdot 2 \sec^2 x \tan x \right) = 2 \left( \sec^4 x + 2 \tan^2 x \sec^2 x \right) $$ 7. **Simplify:** $$ f''(x) = 2 \sec^4 x + 4 \tan^2 x \sec^2 x $$ **Final answer:** $$ f''(x) = 2 \sec^4 x + 4 \tan^2 x \sec^2 x $$