11. $h(x) = \\frac{x^2 - 9}{x^2 + 1}$

Explanation:

Step1: Identify the function

The given function is a rational function: $$h(x) = \frac{x^2 - 9}{x^2 + 1}$$

Step2: Apply the quotient rule

To find the derivative $h'(x)$, use the quotient rule $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$: $$h'(x) = \frac{\frac{d}{dx}(x^2 - 9)(x^2 + 1) - (x^2 - 9)\frac{d}{dx}(x^2 + 1)}{(x^2 + 1)^2}$$

Step3: Differentiate the numerator and denominator

Calculate the derivatives of the individual terms: $$h'(x) = \frac{(2x)(x^2 + 1) - (x^2 - 9)(2x)}{(x^2 + 1)^2}$$

Step4: Simplify the numerator

Distribute $2x$ and combine like terms: $$h'(x) = \frac{2x^3 + 2x - (2x^3 - 18x)}{(x^2 + 1)^2}$$

Step5: Final simplification

Subtract the terms in the numerator: $$h'(x) = \frac{2x^3 + 2x - 2x^3 + 18x}{(x^2 + 1)^2} = \frac{20x}{(x^2 + 1)^2}$$

Answer:

$h'(x) = \frac{20x}{(x^2 + 1)^2}$