# Standard Derivatives

The following list of derivatives of standard functions may be useful. Note that where appropriate $u$ and $v$ are differentiable functions in $x$

### Basic Rules

$$\newcommand{\sech}{\operatorname{sech}} \newcommand{\cosech}{\operatorname{cosech}} \newcommand{\cosec}{\operatorname{cosec}} \newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}} \frac{d}{dx}\left(u^n\right)=nu^{n-1}\frac{du}{dx}$$

$$\frac{d}{dx}\left(c\right)=0,\; \ctext{where c is a constant}$$

$$\frac{d}{dx}\left(u\pm v\right)=\frac{du}{dx}\pm\frac{dv}{dx}$$

$$\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}\qquad\ctext{(Product Rule)}$$

$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}- u\frac{dv}{dx}}{v^2}\qquad\ctext{(Quotient Rule)}$$

$$\ctext{If } y=F(u)\ctext{ where }u=f(x)\ctext{ then }\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$
(Chain Rule or Function of a Function Rule)

### Trigonometric Functions

$$\frac{d}{dx}\left(\sin u\right)=\cos u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cos u\right)=-\sin u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\tan u\right)=\sec^2 u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\sec u\right)=\sec u\tan u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cosec u\right)=-\cosec u\cot u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cot u\right)=-\cosec^2 u\frac{du}{dx}$$

### Inverse Trigonometric Functions

$$\frac{d}{dx}\left(\sin^{-1} u\right)=\frac{1}{\sqrt{1-u^2}}\frac{du}{dx},\qquad -1<u<1$$

$$\frac{d}{dx}\left(\cos^{-1} u\right)=\frac{-1}{\sqrt{1-u^2}}\frac{du}{dx},\qquad -1<u<1$$

$$\frac{d}{dx}\left(\tan^{-1} u\right)=\frac{1}{1+u^2}\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cosec^{-1} u\right)=-\frac{1}{|u|\sqrt{u^2-1}}\frac{du}{dx}\qquad |u|>1$$

$$\frac{d}{dx}\left(\sec^{-1} u\right)=\frac{1}{|u|\sqrt{u^2-1}}\frac{du}{dx}\qquad |u|>1$$

$$\frac{d}{dx}\left(\cot^{-1} u\right)=-\frac{1}{1+u^2}\frac{du}{dx}$$

### Exponential and Logarithmic Functions

$$\frac{d}{dx}\left(e^u\right)=e^u\frac{du}{dx}$$

$$\frac{d}{dx}\left(a^u\right)=a^u\ln a\frac{du}{dx}, \ctext{where }a>0, a\ne 1$$

$$\frac{d}{dx}\left(\ln u\right)=\frac{1}{u}\frac{du}{dx}$$

$$\frac{d}{dx}\left(\ln_a u\right)=\frac{1}{u\ln a}\frac{du}{dx},\ctext{ where }a>0, a\ne 1$$

### Hyperbolic Trig Functions

$$\frac{d}{dx}\left(\sinh u\right)=\cosh u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cosh u\right)=\sinh u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\tanh u\right)=\sech^2 u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\sech u\right)=-\sech u\tanh u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cosech u\right)=-\cosech u\coth u\frac{du}{dx}$$

$$\frac{d}{dx}\left(\coth u\right)=-\cosech^2 u\frac{du}{dx}$$

### Inverse Hyperbolic Trig Functions

$$\frac{d}{dx}\left(\sinh^{-1} u\right)=\frac{1}{\sqrt{1+u^2}}\frac{du}{dx}$$

$$\frac{d}{dx}\left(\cosh^{-1} u\right)=\frac{1}{\sqrt{u^2-1}}\frac{du}{dx},\qquad u>1$$

$$\frac{d}{dx}\left(\tanh^{-1} u\right)=\frac{1}{1-u^2}\frac{du}{dx},\qquad |u|<1$$

$$\frac{d}{dx}\left(\cosech^{-1} u\right)=-\frac{1}{|u|\sqrt{u^2+1}}\frac{du}{dx}\qquad u\ne 0$$

$$\frac{d}{dx}\left(\sech^{-1} u\right)=-\frac{1}{u\sqrt{1-u^2}}\frac{du}{dx}\qquad 0<u<1$$

$$\frac{d}{dx}\left(\coth^{-1} u\right)=-\frac{1}{1-u^2}\frac{du}{dx},\qquad |u|>1$$