Single variable derivatives

Formal definition

Let \(f(x)\) be a function with a single variable \(x\), then the formal definition of the derivative is:

\[ \dfrac{\mathrm{d}f}{\mathrm{d}x} = \lim_{h \rightarrow 0} \dfrac{f(x+h) - f(x)}{h}. \]

Standard derivative table

This table contains the derivatives of the most common functions:

\(f(x)\)	\(f'(x)\)
\(k\)	\(0\)
\(x\)	\(1\)
\(x^k\)	\(kx^{k-1}\)
\(\cos x\)	\(-\sin x\)
\(\sin x\)	\(\cos x\)
\(\tan x\)	\(\sec^2 x\)
\(\cot x\)	\(-\csc^2 x\)
\(\sec x\)	\(\sec x \tan x\)
\(\csc x\)	\(-\csc x \cot x\)
\(\ln x\)	\(\frac{1}{x}\)
\(e^x\)	\(e^x\)
\(k^x\)	\((\ln k)k^x\)