Identify u and v for the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v²).

Let $u = \ln(x)$ and $v = x^2 \sin(x)$.

For $u$: $u' = \frac{1}{x}$.
For $v$, use the product rule: $v' = \frac{d}{dx}(x^2)\sin(x) + x^2\frac{d}{dx}(\sin(x)) = 2x\sin(x) + x^2\cos(x)$.

Factor out $x$ from the numerator.

Cancel one $x$ from numerator and denominator.

4. $y = \frac{\tan(x)}{e^{3x}}$

Let $u = \tan(x)$ and $v = e^{3x}$.

For $u$: $u' = \sec^2(x)$.
For $v$, use the chain rule: $v' = e^{3x} \cdot \frac{d}{dx}(3x) = 3e^{3x}$.

Cancel $e^{3x}$ from numerator and denominator.

6. $f(x) = \frac{2x}{x^2}$

8. $y = \frac{\ln(x^2 + 1)}{\sqrt{x}}$

Let $u = \ln(x^2 + 1)$ and $v = \sqrt{x} = x^{1/2}$.

For $u$, use the chain rule: $u' = \frac{1}{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1) = \frac{2x}{x^2 + 1}$.
For $v$: $v' = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$.

Multiply the numerator and denominator by $2\sqrt{x}(x^2+1)$ to clear fractions in the numerator.

10. $f(x) = \frac{e^x \ln x}{x^2 + 1}$

Let $u = e^x \ln x$ and $v = x^2 + 1$.

For $u$, use the product rule: $u' = \frac{d}{dx}(e^x)\ln x + e^x\frac{d}{dx}(\ln x) = e^x \ln x + e^x \cdot \frac{1}{x} = e^x\left(\ln x + \frac{1}{x}\right)$.
For $v$: $v' = 2x$.

Factor out $e^x$ from the numerator.

Expand the term in the brackets.

Combine like terms.