Derivative Formulas

$\bullet \;\; \displaystyle \frac{d}{dx}x^n=nx^{n-1}$

$\bullet \;\; \displaystyle \frac{d}{dx}a=0$

$\bullet \;\; \displaystyle \frac{d}{dx}af(x)=af'(x)\;\; \left [ here\;\; f'(x)= \frac{df}{dx}\right ]$

$\bullet \;\; \displaystyle \frac{d}{dx}(ax+b)^n=n(ax+b)^{n-1} \frac{d}{dx}(ax+b)$

$\bullet \;\; \displaystyle \frac{d}{dx}(f\pm g)=\frac{df}{dx} \pm \frac{g}{dx}$

$\bullet \;\;Product\;\; rule\;\; \displaystyle \frac{d}{dx}(f\times g)=f.\frac{dg}{dx}+g.\frac{df}{dx}$

$\bullet \;\;Quotient\;\; rule\;\; \displaystyle \frac{d}{dx}(\frac{f}{g})=\displaystyle \frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2}$

$\bullet \;\; \displaystyle \frac{d}{dx}(\sin x)=\cos x$

$\bullet \;\; \displaystyle \frac{d}{dx}(\cos x)=-\sin x$

$\bullet \;\; \displaystyle \frac{d}{dx}(\tan x)=\sec^2 x \;\; where$
$\displaystyle x \neq (2n+1)\frac{\pi}{2}, n\in \mathbb{Z}$

$\bullet \;\; \displaystyle \frac{d}{dx}(\cot x)=-\csc^2 x \;\; where$
$x \neq n\pi \;\;n \in \mathbb{Z}$

$\bullet \;\; \displaystyle \frac{d}{dx}(\sec x)=\sec x\tan x\;\; where$
$\displaystyle x \neq (2n+1)\frac{\pi}{2}, n\in \mathbb{Z}$

$\bullet \;\; \displaystyle \frac{d}{dx}(\csc x)=-\cot x\csc x \;\;\; where$
$x \neq n \pi, n \in \mathbb{Z}$

$\bullet \;\; \displaystyle \frac{d}{dx}(\sin^{-1} x)=\frac{1}{\sqrt{1-x^2}} \;\; where \;\;|x|<1$

$\bullet \;\; \displaystyle \frac{d}{dx}(\cos^{-1} x)=-\frac{1}{\sqrt{1-x^2}}$
$\;\; where\;\; |x|<1$

$\bullet \;\; \displaystyle \frac{d}{dx}(\tan^{-1} x)=\frac{1}{1+x^2}$
$\;\;where\;\;-\infty < x < \infty$

$\bullet \;\; \displaystyle \frac{d}{dx}(\cot^{-1} x)=-\frac{1}{1+x^2}$
$\;\;where\;\;-\infty < x <\infty$

$\bullet \;\; \displaystyle \frac{d}{dx}(\sec^{-1} x)=\frac{1}{|x| \sqrt{x^2-1}} \;\; where\;\; |x|>1$

$\bullet \;\; \displaystyle \frac{d}{dx}(\csc^{-1} x)=-\frac{1}{|x| \sqrt{x^2-1}} \;\;\; where\;\; |x|>1$

$\bullet \;\; \displaystyle \frac{d}{dx}(a^x)=a^x \ln a$

$\bullet \;\; \displaystyle \frac{d}{dx}(e^x)=e^x$

$\bullet \;\; \displaystyle \frac{d}{dx}(e^{ax})=e^{ax}.\frac{d}{dx}(ax)$

$\bullet \;\; \displaystyle \frac{d}{dx}(\log_a x)=\frac{1}{x \ln a}\;\; ,$ $x>0$

$\bullet \;\; \displaystyle \frac{d}{dx}( bx)= b$

$\bullet \;\; \displaystyle \frac{d}{dx}(\sinh x)=\cosh x$

$\bullet \;\; \displaystyle \frac{d}{dx}(\cosh x)=\sinh x$

$\bullet \;\; \displaystyle \frac{d}{dx}(\tanh x)=sech^2 x$