Rules of Differentiation

$f$ and $g$ are differentiable functions of$x$, and $c$ is a constant

Constant Rule

$\frac{d}{dx}(c)=0$

Constant Multiple Rule

$\frac{d}{dx}(c\cdot f(x))=c\cdot f^{\prime }(x)$ $\frac{d}{dx}(cu)=c\frac{du}{dx}$

Sum Rule

$\frac{d}{dx}(f(x)+g(x))=f^{\prime }(x)+g^{\prime }(x)$ $\frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx}$

Product Rule

$\frac{d}{dx}(f(x)g(x))=f^{\prime }(x)g(x)+g^{\prime }(x)f(x)$ $\frac{d}{dx}(uv)=v\frac{du}{dx}+u\frac{dv}{dx}$

Quotient Rule

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime }(x)g(x)-g^{\prime }(x)f(x)}{(g(x){)}^2}$ $\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

Chain Rule

Let $f(u)$ be differentiable at $u=g(x)$, and let $g$ be a differentiable function of $x$ $\frac{d}{dx}(f(g(x)))=f^{\prime }(g(x))\cdot g^{\prime }(x)$

By writing $y=f(g(x))$ and $u=g(x)$, chain rule is $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$ with the understanding that $\frac{dy}{du}$ is evaluated at $u=g(x)$, so that all these expressions are regarded as functions of $x$

Inverse Function

$(f^{-1}{)}^{\prime }(a)=\frac{1}{f^{\prime }(f^{-1}(a))}$