(1) Product rule
$$\frac{dy}{dx} (uv) = v \frac{du}{dx} + u \frac{dv}{dx}$$
Example: \( \frac{dy}{dx} [3x^2 (\ln x)] \)
\( u = 3x^2 \)
\( \frac{du}{dx} = 6x \)
\( v = \ln x \)
\( \frac{dv}{dx} = \frac{1}{x} \)
Substitute into equation:
\( \ln x (6x) + 3x^2 \left(\frac{1}{x}\right) \)
\( = 6x \ln x + 3x \)
(2) Quotient rule
$$\frac{dy}{dx} \left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$
Example: \( \frac{dy}{dx} \left(\frac{3x^2}{\ln x}\right) \)
\( u = 3x^2 \)
\( \frac{du}{dx} = 6x \)
\( v = \ln x \)
\( \frac{dv}{dx} = \frac{1}{x} \)
Substitute into equation:
$$\frac{\ln x (6x) - 3x^2 \left(\frac{1}{x}\right)}{(\ln x)^2}$$
\( = \frac{6x \ln x - 3x}{(\ln x)^2} \)
\( = \frac{6x}{\ln x} - \frac{3x}{(\ln x)^2} \)