Differentiation Formulas — Question 9

Question 9

Let $$f(x) = \tan^2(x) + \sec^2(x)$$

• Differentiate $f(x)$ using the standard differentiation formulas for trigonometric functions.

• Simplify your answer.

Question 9 - Solution

(a) Differentiate $f(x)$:

We are given: $$f(x) = \tan^2(x) + \sec^2(x)$$

We apply the chain rule and known derivatives:

$\frac{d}{dx}[\tan^2(x)] = 2\tan(x)\sec^2(x)$ (using chain rule)

$\frac{d}{dx}[\sec^2(x)] = 2\sec^2(x)\tan(x)$ (using chain rule)

So: $$f'(x) = 2\tan(x)\sec^2(x) + 2\sec^2(x)\tan(x)$$

(b) Simplify:

Both terms are the same: $$f'(x) = 4\tan(x)\sec^2(x)$$

Final Answer: $$\boxed{f'(x) = 4\tan(x)\sec^2(x)}$$