Differentiation techniques

Derivatives of basic functions

Here $\log$ denotes the natural logarithm (base $e$).

Product, quotient and chain rules

Examples: $\{(2x + 1)^5\}' = 5(2x + 1)^4 \cdot 2 = 10(2x + 1)^4$ and $(x^2e^x)' = 2xe^x + x^2e^x = (x^2 + 2x)e^x$.

Logarithmic, implicit and parametric differentiation

Logarithmic differentiation: for $y = x^x$, take logs to get $\log y = x\log x$, differentiate to get $\frac{y'}{y} = \log x + 1$, so $y' = x^x(\log x + 1)$.

Implicit differentiation: for $x^2 + y^2 = 25$, treat $y$ as a function of $x$: $2x + 2y\frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.

Parametric curves: if $x = f(t)$ and $y = g(t)$, then $\displaystyle\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and $\displaystyle\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$.

Higher derivatives

Differentiating $y'$ again gives $y''$; differentiating $n$ times gives $y^{(n)}$. Compute $y'$, $y''$, $y'''$ and look for a pattern.