Calculus Calculus identities. Derivatives ddxc=0\begin{align} \dv{}{x}c = 0 \end{align}dxdc=0 ddx[f(x)+g(x)]=f′(x)+g′(x)\begin{align} \dv{}{x}\left[f(x)+g(x)\right] = f'(x) + g'(x) \end{align}dxd[f(x)+g(x)]=f′(x)+g′(x) ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\begin{align} \dv{}{x}\left[f(x)g(x)\right] = f'(x)g(x) + f(x)g'(x) \end{align}dxd[f(x)g(x)]=f′(x)g(x)+f(x)g′(x) ddxf(x)g(x)=g(x)f′(x)−f(x)g′(x)[g(x)]2\begin{align} \dv{}{x}\frac{f(x)}{g(x)} = \frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2} \end{align}dxdg(x)f(x)=[g(x)]2g(x)f′(x)−f(x)g′(x) ddxf(g(x))=f′(g(x))g′(x)\begin{align} \dv{}{x}f(g(x)) = f'(g(x))g'(x) \end{align}dxdf(g(x))=f′(g(x))g′(x) ddxxn=nxn−1\begin{align} \dv{}{x}x^n = nx^{n-1} \end{align}dxdxn=nxn−1 Exponential and Logarithmic Functions ddxex=ex\begin{align} \dv{}{x} e^x = e^x \end{align}dxdex=ex ddxax=axlna\begin{align} \dv{}{x} a^x = a^x \ln a \end{align}dxdax=axlna ddxln∣x∣=1x\begin{align} \dv{}{x} \ln |x| = \frac{1}{x} \end{align}dxdln∣x∣=x1 ddxlogax=1xlna\begin{align} \dv{}{x} \log_a x = \frac{1}{x \ln a} \end{align}dxdlogax=xlna1 Trigonometric Functions ddxsinx=cosx\begin{align} \dv{}{x} \sin x = \cos x \end{align}dxdsinx=cosx ddxcosx=−sinx\begin{align} \dv{}{x} \cos x = -\sin x \end{align}dxdcosx=−sinx ddxtanx=sec2x\begin{align} \dv{}{x} \tan x = \sec^2 x \end{align}dxdtanx=sec2x ddxcscx=−cscxcotx\begin{align} \dv{}{x} \csc x = -\csc x \cot x \end{align}dxdcscx=−cscxcotx ddxsecx=secxtanx\begin{align} \dv{}{x} \sec x = \sec x \tan x \end{align}dxdsecx=secxtanx ddxcotx=−csc2x\begin{align} \dv{}{x} \cot x = -\csc^2 x \end{align}dxdcotx=−csc2x