Common derivatives, with the why

Each page gives the derivative up front, then proves it, shows the mistakes students actually make, and links to practice. No paywall in front of the steps.

$\dfrac{d}{dx}\left[\ln x\right] = \frac{1}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[e^x\right] = e^x$ proof and mistakes
$\dfrac{d}{dx}\left[\sin x\right] = \cos x$ proof and mistakes
$\dfrac{d}{dx}\left[\cos x\right] = -\sin x$ proof and mistakes
$\dfrac{d}{dx}\left[\tan x\right] = \sec^2 x$ proof and mistakes
$\dfrac{d}{dx}\left[\sec x\right] = \sec x \tan x$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{x}\right] = \frac{1}{2\sqrt{x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{x}\right] = -\frac{1}{x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[a^x\right] = a^x \ln a$ proof and mistakes
$\dfrac{d}{dx}\left[x^x\right] = x^x(\ln x + 1)$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan x\right] = \frac{1}{1+x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[\arcsin x\right] = \frac{1}{\sqrt{1-x^2}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{2}\right] = 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\cot x\right] = -\csc^{2} x$ proof and mistakes
$\dfrac{d}{dx}\left[\csc x\right] = -\csc x\cot x$ proof and mistakes
$\dfrac{d}{dx}\left[\sin 2x\right] = 2\cos 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\cos 2x\right] = -2\sin 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\sin^{2} x\right] = 2\sin x\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[\cos^{2} x\right] = -2\sin x\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[\tan^{2} x\right] = 2\tan x\sec^{2} x$ proof and mistakes
$\dfrac{d}{dx}\left[e^{2x}\right] = 2e^{2x}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln 2x\right] = \frac{1}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[\log_{10} x\right] = \frac{1}{x\ln 10}$ proof and mistakes
$\dfrac{d}{dx}\left[\arccos x\right] = -\frac{1}{\sqrt{1 - x^{2}}}$ proof and mistakes
$\dfrac{d}{dx}\left[x\sin x\right] = \sin x + x\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[x\ln x\right] = \ln x + 1$ proof and mistakes
$\dfrac{d}{dx}\left[e^{x^{2}}\right] = 2xe^{x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{x^{2}}\right] = -\frac{2}{x^{3}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^3\right] = 3x^2$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{\sqrt{x}}\right] = -\frac{1}{2x\sqrt{x}}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{3x}\right] = 3e^{3x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sin 3x\right] = 3\cos 3x$ proof and mistakes
$\dfrac{d}{dx}\left[\cos 3x\right] = -3\sin 3x$ proof and mistakes
$\dfrac{d}{dx}\left[x\cos x\right] = \cos x - x\sin x$ proof and mistakes
$\dfrac{d}{dx}\left[xe^x\right] = e^x(x+1)$ proof and mistakes
$\dfrac{d}{dx}\left[\sin x\cos x\right] = \cos 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\ln(x^2)\right] = \frac{2}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{\sin x}\right] = \cos x\,e^{\sin x}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln(\cos x)\right] = -\tan x$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{x^2+1}\right] = \frac{x}{\sqrt{x^2+1}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{x+1}\right] = \frac{1}{(x+1)^2}$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan 2x\right] = \frac{2}{1+4x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[x\tan x\right] = \tan x + x\sec^2 x$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{x^3}\right] = -\frac{3}{x^4}$ proof and mistakes
$\dfrac{d}{dx}\left[x^4\right] = 4x^3$ proof and mistakes
$\dfrac{d}{dx}\left[e^{-x}\right] = -e^{-x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sin(x^2)\right] = 2x\cos(x^2)$ proof and mistakes
$\dfrac{d}{dx}\left[\cos(x^2)\right] = -2x\sin(x^2)$ proof and mistakes
$\dfrac{d}{dx}\left[x^2 e^x\right] = e^x(x^2+2x)$ proof and mistakes
$\dfrac{d}{dx}\left[x^2\sin x\right] = 2x\sin x + x^2\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[e^x\cos x\right] = e^x(\cos x-\sin x)$ proof and mistakes
$\dfrac{d}{dx}\left[e^x\sin x\right] = e^x(\sin x+\cos x)$ proof and mistakes
$\dfrac{d}{dx}\left[\tan 2x\right] = 2\sec^2 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\sec 2x\right] = 2\sec 2x\tan 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\ln 3x\right] = \frac{1}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt[3]{x}\right] = \frac{1}{3\sqrt[3]{x^2}}$ proof and mistakes
$\dfrac{d}{dx}\left[(x^2+1)^3\right] = 6x(x^2+1)^2$ proof and mistakes
$\dfrac{d}{dx}\left[\sin^3 x\right] = 3\sin^2 x\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[\arcsin 2x\right] = \frac{2}{\sqrt{1-4x^2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\cos^3 x\right] = -3\sin x\cos^2 x$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{\ln x}\right] = \frac{\ln x-1}{(\ln x)^2}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\sin x}{x}\right] = \frac{x\cos x-\sin x}{x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{e^x}{x}\right] = \frac{e^x(x-1)}{x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln(\sin x)\right] = \cot x$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{\sin x}\right] = \frac{\cos x}{2\sqrt{\sin x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan(x^2)\right] = \frac{2x}{1+x^4}$ proof and mistakes
$\dfrac{d}{dx}\left[x\arctan x\right] = \arctan x+\frac{x}{1+x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[x\arcsin x\right] = \arcsin x+\frac{x}{\sqrt{1-x^2}}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{2x}\sin x\right] = e^{2x}(2\sin x+\cos x)$ proof and mistakes
$\dfrac{d}{dx}\left[\tan(x^2)\right] = 2x\sec^2(x^2)$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\ln x}{x}\right] = \frac{1-\ln x}{x^2}$ proof and mistakes
$\dfrac{d}{dx}\left[x^2\ln x\right] = x(2\ln x+1)$ proof and mistakes
$\dfrac{d}{dx}\left[\sec^2 x\right] = 2\sec^2 x\tan x$ proof and mistakes
$\dfrac{d}{dx}\left[3^x\right] = 3^x\ln 3$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{1-x^2}\right] = \frac{-x}{\sqrt{1-x^2}}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{\cos x}\right] = -\sin x\,e^{\cos x}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{x^{2}+1}\right] = \frac{1-x^{2}}{\left(x^{2}+1\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x^{2}}{x+1}\right] = \frac{x^{2}+2x}{\left(x+1\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{x+1}\right] = \frac{1}{2\sqrt{x+1}}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln(x+1)\right] = \frac{1}{x+1}$ proof and mistakes
$\dfrac{d}{dx}\left[\cot 2x\right] = -2\csc^{2} 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\csc 2x\right] = -2\csc 2x\cot 2x$ proof and mistakes
$\dfrac{d}{dx}\left[\sin 4x\right] = 4\cos 4x$ proof and mistakes
$\dfrac{d}{dx}\left[\tan 3x\right] = 3\sec^{2} 3x$ proof and mistakes
$\dfrac{d}{dx}\left[x^{3}e^{x}\right] = x^{2}e^{x}(3+x)$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{e^{x}}\right] = \frac{1-x}{e^{x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln\left(\ln x\right)\right] = \frac{1}{x\ln x}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{\tan x}\right] = \sec^{2}x\,e^{\tan x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sin\left(e^{x}\right)\right] = e^{x}\cos\left(e^{x}\right)$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{\tan x}\right] = \frac{\sec^{2}x}{2\sqrt{\tan x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{\ln x}\right] = -\frac{1}{x\left(\ln x\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{5}\right] = 5x^{4}$ proof and mistakes
$\dfrac{d}{dx}\left[2^{x}\right] = 2^{x}\ln 2$ proof and mistakes
$\dfrac{d}{dx}\left[\cos 4x\right] = -4\sin 4x$ proof and mistakes
$\dfrac{d}{dx}\left[x^{2}\cos x\right] = 2x\cos x - x^{2}\sin x$ proof and mistakes
$\dfrac{d}{dx}\left[\ln\left(x^{3}\right)\right] = \frac{3}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sec x \tan x\right] = \sec x\left(2\sec^{2}x - 1\right)$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{\sin x}\right] = \frac{\sin x - x\cos x}{\sin^{2}x}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln\left(\tan x\right)\right] = \frac{1}{\sin x\cos x}$ proof and mistakes
$\dfrac{d}{dx}\left[\arcsin\left(x^{2}\right)\right] = \frac{2x}{\sqrt{1-x^{4}}}$ proof and mistakes
$\dfrac{d}{dx}\left[\arccos\left(2x\right)\right] = -\frac{2}{\sqrt{1-4x^{2}}}$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan\left(\frac{1}{x}\right)\right] = -\frac{1}{1+x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{\sin x}\right] = x^{\sin x}\left(\cos x\ln x + \frac{\sin x}{x}\right)$ proof and mistakes
$\dfrac{d}{dx}\left[x^{\ln x}\right] = \frac{2\ln x \cdot x^{\ln x}}{x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sqrt{\ln x}\right] = \frac{1}{2x\sqrt{\ln x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln\sqrt{x}\right] = \frac{1}{2x}$ proof and mistakes
$\dfrac{d}{dx}\left[\sin\sqrt{x}\right] = \frac{\cos\sqrt{x}}{2\sqrt{x}}$ proof and mistakes
$\dfrac{d}{dx}\left[e^{\sqrt{x}}\right] = \frac{e^{\sqrt{x}}}{2\sqrt{x}}$ proof and mistakes
$\dfrac{d}{dx}\left[\cos\left(\frac{1}{x}\right)\right] = \frac{\sin\left(\frac{1}{x}\right)}{x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\ln\left(x+\sqrt{x^{2}+1}\right)\right] = \frac{1}{\sqrt{x^{2}+1}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{1}{1+e^{x}}\right] = -\frac{e^{x}}{\left(1+e^{x}\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{e^{x}}{1+e^{x}}\right] = \frac{e^{x}}{\left(1+e^{x}\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x}{\sqrt{x^{2}+1}}\right] = \frac{1}{\left(x^{2}+1\right)^{3/2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\sec^{3}x\right] = 3\sec^{3}x\tan x$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{x^{2}-1}{x^{2}+1}\right] = \frac{4x}{\left(x^{2}+1\right)^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\sin x}{1+\cos x}\right] = \frac{1}{1+\cos x}$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan\left(e^{x}\right)\right] = \frac{e^{x}}{1+e^{2x}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{6}\right] = 6x^{5}$ proof and mistakes
$\dfrac{d}{dx}\left[\csc^{2} x\right] = -2\csc^{2}x\cot x$ proof and mistakes
$\dfrac{d}{dx}\left[\operatorname{arccot} x\right] = -\frac{1}{1+x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{3}\sin x\right] = 3x^{2}\sin x + x^{3}\cos x$ proof and mistakes
$\dfrac{d}{dx}\left[x\arccos x\right] = \arccos x - \frac{x}{\sqrt{1-x^{2}}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\sin x}{x^{2}}\right] = \frac{x\cos x - 2\sin x}{x^{3}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\cos x}{x}\right] = \frac{-x\sin x - \cos x}{x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\tan x}{x}\right] = \frac{x\sec^{2}x - \tan x}{x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[x^{2}\arctan x\right] = 2x\arctan x + \frac{x^{2}}{1+x^{2}}$ proof and mistakes
$\dfrac{d}{dx}\left[\frac{\ln x}{x^{2}}\right] = \frac{1 - 2\ln x}{x^{3}}$ proof and mistakes
$\dfrac{d}{dx}\left[\arctan\sqrt{x}\right] = \frac{1}{2\sqrt{x}\left(1+x\right)}$ proof and mistakes
$\dfrac{d}{dx}\left[\sec\left(x^{2}\right)\right] = 2x\sec\left(x^{2}\right)\tan\left(x^{2}\right)$ proof and mistakes