Formulas for working with derivatives

Some useful formulas for calculating derivatives of functions.

Basic formulas

Basic formulas that you will use in almost any calculation of the derivative of a function. The first column is the original function, the second column is the derivative of the function. We assume that we are deriving by x and that c is a constant.

$$\begin{eqnarray} c^\prime&=&0\\ x^\prime&=&1\\ (x^c)^\prime&=&cx^{c-1} \end{eqnarray}$$

Addition, multiplication and division

Assume that f(x) and f and g(x) and g are some functions, respectively. Then we can write:

$$\begin{eqnarray} (f+g)^\prime&=&f^\prime+g^\prime\\ (c\cdot f)^\prime&=&c\cdot f^\prime\\ (f\cdot g)^\prime&=&f^\prime\cdot g+f\cdot g^\prime\\ \left(\frac{f}{g}\right)^\prime&=&\frac{f^\prime\cdot g-f\cdot g^\prime}{g^2}\\ \end{eqnarray}$$

Specially then we have the derivative of a composite function:

$$\begin{eqnarray} (f(g(x)))^\prime&=&f^\prime(g(x))\cdot g^\prime(x) \end{eqnarray}$$

Logarithms and exponential functions

$$\begin{eqnarray} (c^x)^\prime&=&c^x\ln c;\quad c>0\\ (e^x)^\prime&=&e^x\\ (\log_cx)^\prime&=&\frac{1}{x\cdot \ln c};\quad c>0\wedge c\ne0\\ (\ln x)^\prime&=&\frac{1}{x} \end{eqnarray}$$

Goniometric functions

$$\begin{eqnarray} (\sin x)^\prime&=&\cos x\\ (\cos x)^\prime&=&-\sin x\\ (\tan x)^\prime&=&\frac{1}{\cos^2x}\\ (\mbox{cotan},x)^\prime&=&-\frac{1}{\sin^2x}\\ (\arcsin x)^\prime&=&\frac{1}{\sqrt{1-x^2}}\\ (\arccos x)^\prime&=&-\frac{1}{\sqrt{1-x^2}}\\ (\arctan x)^\prime&=&\frac{1}{1+x^2}\\ (\mbox{arccotan}, x)^\prime&=&-\frac{1}{1+x^2}\\ \end{eqnarray}$$