Exponential and log derivatives

Derivative of ln x \frac{d}{dx}\,\ln x = \frac{1}{x} The derivative of ln x is 1/x. See why, with a proof from eʸ = x, the chain rule version for ln(u), a graph and worked examples like ln(2x) and ln(x²+1). Derivative of e^x \frac{d}{dx}\,e^x = e^x The derivative of eˣ is eˣ. See why with a limit proof, the chain rule form for e^u, a graph and worked examples like e^(3x) and x²eˣ. Derivative of e^(2x) \frac{d}{dx}\,e^{2x} = 2e^{2x} The derivative of e^(2x) is 2e^(2x). See the chain rule steps, a second proof using (eˣ)², higher derivatives, a graph and worked examples. Derivative of e^(−x) \frac{d}{dx}\,e^{-x} = -e^{-x} The derivative of e^(−x) is −e^(−x). See the chain rule steps, a check with the quotient rule, higher derivatives, a graph and examples like xe^(−x). Derivative of e^(x²) \frac{d}{dx}\,e^{x^2} = 2x\,e^{x^2} The derivative of e^(x²) is 2x·e^(x²). See the chain rule steps, the second derivative, how it differs from (eˣ)², and examples like e^(−x²). Derivative of 2^x \frac{d}{dx}\,2^x = 2^x\ln 2 The derivative of 2ˣ is 2ˣ ln 2, and of aˣ is aˣ ln a. See the proof by rewriting aˣ as e^(x ln a), chain rule examples and a graph. Derivative of log x \frac{d}{dx}\,\log_{10} x = \frac{1}{x\ln 10} The derivative of log₁₀ x is 1/(x ln 10), and of log_b x is 1/(x ln b). See the change of base proof, chain rule examples and a graph. Derivative of ln(2x) \frac{d}{dx}\,\ln(2x) = \frac{1}{x} The derivative of ln(2x) is 1/x, the same as ln x. See two proofs, a graph of why the slopes match, and how ln(2x + 1) is different. Derivative of ln(x²) \frac{d}{dx}\,\ln(x^2) = \frac{2}{x} The derivative of ln(x²) is 2/x, while the derivative of (ln x)² is 2 ln x / x. See both proofs, a graph and how to tell them apart. Derivative of x ln x \frac{d}{dx}\,x\ln x = \ln x + 1 The derivative of x ln x is ln x + 1. See the product rule steps, the minimum at x = 1/e, the second derivative, a graph and similar examples. Derivative of x e^x \frac{d}{dx}\,xe^x = e^x(x + 1) The derivative of xeˣ is eˣ(x + 1). See the product rule steps, the minimum at x = −1, the nth derivative pattern, a graph and worked examples. Derivative of x^x \frac{d}{dx}\,x^x = x^x(\ln x + 1) The derivative of xˣ is xˣ(ln x + 1). Learn two methods, logarithmic differentiation and rewriting as e^(x ln x), plus a graph and examples. Derivative of ln(cos x) \frac{d}{dx}\,\ln(\cos x) = -\tan x The derivative of ln(cos x) is −tan x. See the chain rule steps, why it explains the integral of tan x, a graph, and ln(sin x) and ln(sec x).