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