Derivative Rules

$\frac{d}{dx}(x^n) = nx^{n-1}$, $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$

$\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(\ln x) = \frac{1}{x}$, $\frac{d}{dx}(\tan x) = \sec^2 x$

Chain Rule

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Product Rule

$(uv)' = u'v + uv'$

Quotient Rule

$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Chain: $\frac{d}{dx}(\sin(3x^2)) = \cos(3x^2) \cdot 6x$.

Product: $\frac{d}{dx}(x^2 e^x) = 2xe^x + x^2 e^x$.

Quotient: $\frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{\frac{1}{x} \cdot x - \ln x}{x^2} = \frac{1 - \ln x}{x^2}$.