Differentiation of functions

Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x

Summary of results discussed so far at a glance

$\displaystyle \frac { d }{ dx } \left( { x }^{ n } \right) =n{ x }^{ n-1 }$
$\displaystyle \frac { d }{ dx } \left( { e }^{ x } \right) ={ e }^{ x }$
$\displaystyle \frac { d }{ dx } \left( { a }^{ x } \right) ={ a }^{ x }\log _{ e }{ a }$
$\displaystyle \frac { d }{ dx } \left( \log _{ e }{ x } \right) =\frac { 1 }{ x }$
$\displaystyle \frac { d }{ dx } \left( \log _{ a }{ x } \right) =\frac { 1 }{ x\log _{ e }{ a } }$
$\displaystyle \frac { d }{ dx } \left( \sin { x } \right) =\cos { x }$
$\displaystyle \frac { d }{ dx } \left( \cos { x } \right) =-\sin { x }$
$\displaystyle \frac { d }{ dx } \left( \tan { x } \right) =\sec ^{ 2 }{ x }$
$\displaystyle \frac { d }{ dx } \left( \cot { x } \right) =-{ cosec }^{ 2 }x$
$\displaystyle \frac { d }{ dx } \left( \sec { x } \right) =\sec { x } \tan { x }$
$\displaystyle \frac { d }{ dx } \left( cosec\quad x \right) =-cosec\quad x\quad \cot { x }$
$\displaystyle \frac { d }{ dx } \left( \sin ^{ -1 }{ x } \right) =\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$
$\displaystyle \frac { d }{ dx } \left( \cos ^{ -1 }{ x } \right) =-\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$
$\displaystyle \frac { d }{ dx } \left( \tan ^{ -1 }{ x } \right) =\frac { 1 }{ 1+{ x }^{ 2 } }$
$\displaystyle \frac { d }{ dx } \left( \cot ^{ -1 }{ x } \right) =-\frac { 1 }{ 1+{ x }^{ 2 } }$
$\displaystyle \frac { d }{ dx } \left( \sec ^{ -1 }{ x } \right) =\frac { 1 }{ \left| x \right| \sqrt { { x }^{ 2 }-1 } }$
$\displaystyle \frac { d }{ dx } \left( { cosec }^{ -1 }x \right) =-\frac { 1 }{ \left| x \right| \sqrt { { x }^{ 2 }-1 } }$