Derivative rules: Products and quotients

Find the following derivatives. Indicate which derivative rules are being used.

1. Let \( y = x^2 \cdot \ln{x} \). What is \( \frac{dy}{dx} \)? (Recall that \( \frac{d}{dx} \ln{x} = \frac{1}{x} \).)


2. Suppose \( g(x) = \dfrac{x^2}{x^9+1} \). What is the derivative of \( g(x) \) with respect to \( x \)?


Challenge

3. Find \( \frac{d}{dx} (x^2 \cdot \ln{x} \cdot e^x) \).


4. Let \( h(x) = 2x^3 \cdot \ln{x} \). What is \( h''(x) \)?