Derivative Rules

Primary

• $\frac{d}{dx}[c]=0$
• $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$
• $\frac{d}{dx}[x^n]=n\cdot x^{n-1}$
• $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$
• $\frac{d}{dx}[f(x)\cdot g(x)]=f(x)g(x)+g(x)f(x)$
• $\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f^{\prime }(x)-f(x)g\prime (x)}{[g(x){]}^2}$
• $\frac{d}{dx}[e^x]=e^x$
• $\frac{d}{dx}[a^x]=a^x\cdot \ln a$
• $\frac{d}{dx}[\ln x]=\frac{1}{x}$
• $\frac{d}{dx}[{\log }_{a}x]=\frac{1}{x\ln a}$
• $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$
• $\frac{d}{dx}[f^{-1}(x)]=\frac{1}{f^{\prime }(f^{-1}(x))}$
• $\frac{d}{dx}[f^{\prime }(x)]=f^{\prime \prime }(x)$

Trig

• $\frac{d}{dx}[\sin x]=\cos x$
• $\frac{d}{dx}[\cos x]=-\sin x$
• $\frac{d}{dx}[\tan x]={\sec }^{2}x$
• $\frac{d}{dx}[\sec x]=\sec x\cdot \tan x$
• $\frac{d}{dx}[\cot x]=-{\csc }^{2}x$
• $\frac{d}{dx}[\csc x]=-\csc x\cdot \cot x$
• $\frac{d}{dx}[{\sin }^{-1}x]=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[{\tan }^{-1}x]=\frac{1}{1+x^2}$
• $\frac{d}{dx}[{\sec }^{-1}x]=\frac{1}{|x|\sqrt{x^2-1}}$