$\frac{d}{d x}(k)=0$ where $k$ is a constant $\frac{d}{d x}(f(x)+g(x))=f^{\prime}(x)+g^{\prime}(x)$ $\frac{d}{d x}(f(x) g(x))=f^{\prime}(x) g(x)+f(x) g^{\prime}(x)$ $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$, for real numbers $n$ $\frac{d}{d x}(c f(x))=c f^{\prime}(x)$ $\frac{d}{d x}(f(x)-g(x))=f^{\prime}(x)-g^{\prime}(x)$ $\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{(g(x))^{2}}$ $\frac{d}{d x}[f(g(x))]=f^{\prime}(g(x)) \cdot g^{\prime}(x)$ $\frac{d}{d x}(\sin x)=\cos x$ $\frac{d}{d x}(\tan x)=\sec ^{2} x$ $\frac{d}{d x}(\sec x)=\sec x \tan x$ $\frac{d}{d x}(\cos x)=-\sin x$ $\frac{d}{d x}(\cot x)=-\csc ^{2} x$ $\frac{d}{d x}(\csc x)=-\csc x \cot x$ $\frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}$ $\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^{2}}$ $\frac{d}{d x}\left(\sec ^{-1} x\right)=\frac{1}{|x| \sqrt{x^{2}-1}}$ $\frac{d}{d x}\left(\cos ^{-1} x\right)=-\frac{1}{\sqrt{1-x^{2}}}$ $\frac{d}{d x}\left(\cot ^{-1} x\right)=-\frac{1}{1+x^{2}}$ $\frac{d}{d x}\left(\csc ^{-1} x\right)=-\frac{1}{|x| \sqrt{x^{2}-1}}$ $\frac{d}{d x}\left(e^{x}\right)=e^{x}$ $\frac{d}{d x}(\ln |x|)=\frac{1}{x}$ $\frac{d}{d x}\left(b^{x}\right)=b^{x} \ln b$ $\frac{d}{d x}\left(\log _{b} x\right)=\frac{1}{x \ln b}$ $\frac{d}{d x}(\sinh x)=\cosh x$ $\frac{d}{d x}(\tanh x)=\operatorname{sech}^{2} x$ $\frac{d}{d x}(\operatorname{sech} x)=-\operatorname{sech} x \tanh x$ $\frac{d}{d x}(\cosh x)=\sinh x$ $\frac{d}{d x}(\operatorname{coth} x)=-\operatorname{csch}^{2} x$ $\frac{d}{d x}(\operatorname{csch} x)=-\operatorname{csch} x \operatorname{coth} x$ $\begin{aligned} \frac{d}{d x}\left(\sinh ^{-1} x\right) &=\frac{1}{\sqrt{x^{2}+1}} \\ \frac{d}{d x}\left(\tanh ^{-1} x\right) &=\frac{1}{1-x^{2}}(|x|<1) \\ \frac{d}{d x}\left(\operatorname{sech}^{-1} x\right) &=-\frac{1}{x \sqrt{1-x^{2}}} \quad(0<x<1) \\ \frac{d}{d x}\left(\cosh ^{-1} x\right) &=\frac{1}{\sqrt{x^{2}-1}} \quad(x>1) \\ \frac{d}{d x}\left(\operatorname{coth}^{-1} x\right) &=\frac{1}{1-x^{2}} \quad(|x|>1) \\ \frac{d}{d x}\left(\operatorname{csch}^{-1} x\right) &=-\frac{1}{|x| \sqrt{1+x^{2}}}(x \neq 0) \end{aligned}$