\(\bullet \;\; \displaystyle \frac{d}{dx}x^n=nx^{n-1}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}a=0\)
\(\bullet \;\; \displaystyle \frac{d}{dx}af(x)=af'(x)\;\; \left [ here\;\; f'(x)= \frac{df}{dx}\right ]\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(ax+b)^n=n(ax+b)^{n-1} \frac{d}{dx}(ax+b)\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(f\pm g)=\frac{df}{dx} \pm \frac{g}{dx}\)
\(\bullet \;\;Product\;\; rule\;\; \displaystyle \frac{d}{dx}(f\times g)=f.\frac{dg}{dx}+g.\frac{df}{dx}\)
\(\bullet \;\;Quotient\;\; rule\;\; \displaystyle \frac{d}{dx}(\frac{f}{g})=\displaystyle \frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\sin x)=\cos x\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\cos x)=-\sin x\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\tan x)=\sec^2 x \;\; where \)
\(\displaystyle x \neq (2n+1)\frac{\pi}{2}, n\in \mathbb{Z}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\cot x)=-\csc^2 x \;\; where \)
\(x \neq n\pi \;\;n \in \mathbb{Z}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\sec x)=\sec x\tan x\;\; where \)
\(\displaystyle x \neq (2n+1)\frac{\pi}{2}, n\in \mathbb{Z}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\csc x)=-\cot x\csc x \;\;\; where \)
\( x \neq n \pi, n \in \mathbb{Z}\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\sin^{-1} x)=\frac{1}{\sqrt{1-x^2}} \;\; where \;\;|x|<1 \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\cos^{-1} x)=-\frac{1}{\sqrt{1-x^2}}\)
\(\;\; where\;\; |x|<1 \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\tan^{-1} x)=\frac{1}{1+x^2}\)
\(\;\;where\;\;-\infty < x < \infty \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\cot^{-1} x)=-\frac{1}{1+x^2}\)
\(\;\;where\;\;-\infty < x <\infty\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\sec^{-1} x)=\frac{1}{|x| \sqrt{x^2-1}} \;\; where\;\;
|x|>1\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\csc^{-1} x)=-\frac{1}{|x| \sqrt{x^2-1}} \;\;\; where\;\;
|x|>1\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(a^x)=a^x \ln a\)
\(\bullet \;\; \displaystyle \frac{d}{dx}(e^x)=e^x \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(e^{ax})=e^{ax}.\frac{d}{dx}(ax) \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\log_a x)=\frac{1}{x \ln a}\;\; ,\) \(x>0\)
\(\bullet \;\; \displaystyle \frac{d}{dx}( bx)= b \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\sinh x)=\cosh x \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\cosh x)=\sinh x \)
\(\bullet \;\; \displaystyle \frac{d}{dx}(\tanh x)=sech^2 x \)
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