Differentiate with respect to x: ((1+x2))/(x2)

let's solve questions 3, 4, and 5.

Question 3

a) Differentiate with respect to x: $\frac{\ln(1+x^2)}{x^2}$

Step 1: Identify the function as a quotient and define $u$ and $v$.
Let $f(x) = \frac{u}{v}$, where $u = \ln(1+x^2)$ and $v = x^2$.

Step 2: Find the derivatives of $u$ and $v$.
Using the chain rule for $u$:

$$\frac{du}{dx} = \frac{1}{1+x^2} \cdot \frac{d}{dx}(1+x^2) = \frac{2x}{1+x^2}$$

Using the power rule for $v$:

$$\frac{dv}{dx} = 2x$$

Step 3: Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.

$$f'(x) = \frac{x^2\left(\frac{2x}{1+x^2}\right) - \ln(1+x^2)(2x)}{(x^2)^2}$$ $$f'(x) = \frac{\frac{2x^3}{1+x^2} - 2x\ln(1+x^2)}{x^4}$$

Step 4: Simplify the expression.
Factor out $2x$ from the numerator:

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