Derivative formulas

The derivative of any polynomial function of one variable is easily obtained. If $c\in \mathbb {R}$ (or a constant function) and $f,g:D\to \mathbb {R}$ are both differentiable on some set $D'$, then so are $c\cdot f$, $f+g$, and $f\cdot g$. If, in addition, $g$ is nonzero on $D'$, then ${\frac {1}{g}}$ (and also ${\frac {f}{g}}$) are differentiable on $D'$. Also, if $f$ is differentiable on $g\left(D'\right)$, then $f\circ g$ is differentiable on $D'$. For the trivial case of $f(x)=a$, for some constant $a$ (a degree 0 polynomial):

$a'=0$[Proof]

For any$r\in \mathbb {R}$:

$(x^{r})'=rx^{r-1}$[Proof]

Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.

For any real-valued differentiable functions $f(x),g(x)$:

• The definition of the derivative:

$f'(x)=\lim _{h\rightarrow 0}{\dfrac {f(x+h)-f(x)}{h}}$

Basic derivative rules:

• $(a\cdot f(x))'=a\cdot f'(x)$[Proof]

• $(f(x)\pm g(x))'=f'(x)\pm g'(x)$[Proof]

• $(-f(x))'=-f'(x)$(Derived from Constant Multiple rule)

• $(f(x)\cdot g(x))'=f'(x)g(x)+f(x)g'(x)$(Product rule)

• $\left({\frac {f(x)}{g(x)}}\right)'={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}$(Quotient rule)

• $\left({\frac {1}{g(x)}}\right)'=-{\frac {g'(x)}{g(x)^{2}}}$(Derived from Quotient rule)

• $(f\circ g)'={\bigl (}f(g(x)){\bigr )}'=(f'(g))\cdot g'(x)$(Chain rule)

Trigonometric functions:

• ${\frac {d}{dx}}(\sin(x))=\cos(x)$[Proof]

• ${\frac {d}{dx}}(\cos(x))=-\sin(x)$

• ${\frac {d}{dx}}(\tan(x))=\sec ^{2}(x)$

• ${\frac {d}{dx}}(\csc(x))=-\csc(x)\cot(x)$

• ${\frac {d}{dx}}(\sec(x))=\sec(x)\tan(x)$

• ${\frac {d}{dx}}(\cot(x))=-\csc ^{2}(x)$

• ${\frac {d}{dx}}(\arcsin(x))={\frac {1}{\sqrt {1-x^{2}}}}$[Proof]

• ${\frac {d}{dx}}(\arccos(x))=-{\frac {1}{\sqrt {1-x^{2}}}}$

• ${\frac {d}{dx}}(\arctan(x))={\frac {1}{1+x^{2}}}$

• ${\frac {d}{dx}}(\operatorname {arcsec}(x))={\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

• ${\frac {d}{dx}}(\operatorname {arccsc}(x))=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

• ${\frac {d}{dx}}(\operatorname {arccot}(x))=-{\frac {1}{1+x^{2}}}$

Logarithmic and exponential functions:

• ${\frac {d}{dx}}(e^{x})=e^{x}$[Proof]

• ${\frac {d}{dx}}(a^{x})=a^{x}\ln(a)$[Proof]

• ${\frac {d}{dx}}(\ln(|x|))={\frac {1}{x}}$[Proof]

• ${\frac {d}{dx}}(\log _{a}(x))={\frac {1}{\ln(a)x}}$[Proof]

Other:

$(f^{-1}(x))'={\frac {1}{f'(f^{-1}(x))}}$[Proof]

${\frac {d}{dx}}(f(x)^{g(x)})={\Big (}g'(x)\ln(f(x))+{\frac {g(x)f'(x)}{f(x)}}{\Big )}f(x)^{g(x)}$[Proof]