\[ f\left(x\right)=c\rightarrow f’\left(x\right)=0 \] \[ f\left(x\right)=x\rightarrow f’\left(x\right)=1 \] \[ f\left(x\right)=x^{b}\rightarrow f’\left(x\right)=bx^{b-1}\;,\; b\in\mathbb{R} \] \[ f\left(x\right)=\sqrt{x}\rightarrow f’\left(x\right)=\frac{1}{2\sqrt{x}} \] \[ f\left(x\right)=\sqrt[n]{x}\rightarrow f’\left(x\right)=\frac{1}{n\sqrt[n]{x^{n-1}}} \] \[ f\left(x\right)=\sin x\rightarrow f’\left(x\right)=\cos x \] \[ f\left(x\right)=\cos x\rightarrow f’\left(x\right)=-\sin x \] \[ f\left(x\right)=\tan x\rightarrow f’\left(x\right)=1+\tan^{2}x=\frac{1}{\cos^{2}x} \] \[ f\left(x\right)=\arcsin x\rightarrow f’\left(x\right)=\frac{1}{\sqrt{1-x^{2}}} \] \[ f\left(x\right)=\arccos x\rightarrow f’\left(x\right)=-\frac{1}{\sqrt{1-x^{2}}} \] \[ f\left(x\right)=\arctan x\rightarrow f’\left(x\right)=\frac{1}{1+x^{2}} \] \[ f\left(x\right)=a^{x}\rightarrow f’\left(x\right)=a^{x}\ln a \] \[ f\left(x\right)=e^{x}\rightarrow f’\left(x\right)=e^{x} \] \[ f\left(x\right)=\log_{a}x\rightarrow f’\left(x\right)=\frac{1}{x}\log_{a}e=\frac{1}{x\ln a} \] \[ f\left(x\right)=\ln x\rightarrow f’\left(x\right)=\frac{1}{x} \]