Differentiation Rules

Chain Rule

$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

$y = (3x+1)^5$: let $u = 3x+1$. $\frac{dy}{du} = 5u^4$, $\frac{du}{dx} = 3$. $\frac{dy}{dx} = 15(3x+1)^4$.

Product Rule

$\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$

$y = x^2\sin x$: $\frac{dy}{dx} = x^2\cos x + 2x\sin x$.

Quotient Rule

$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

$y = \frac{x^2}{\cos x}$: $\frac{dy}{dx} = \frac{2x\cos x + x^2\sin x}{\cos^2 x}$.