The derivative of every common function

Derivative of x \frac{d}{dx}\,x = 1 The derivative of x is 1. See why with the limit definition and the power rule, plus derivatives of 5x, ax + b and x with respect to other variables. Derivative of x² \frac{d}{dx}\,x^2 = 2x The derivative of x² is 2x. See the limit definition proof, the chain rule version for u², a graph, tangent lines and worked examples like (3x + 1)². Derivative of √x \frac{d}{dx}\,\sqrt{x} = \frac{1}{2\sqrt{x}} The derivative of √x is 1/(2√x). See the power rule and limit proofs, the chain rule for √u, why x = 0 fails, and examples like √(x² + 1). Derivative of ∛x \frac{d}{dx}\,\sqrt[3]{x} = \frac{1}{3\sqrt[3]{x^2}} The derivative of the cube root of x is 1/(3∛(x²)). See the power rule and limit proofs, the vertical tangent at 0 and chain rule examples. Derivative of 1/x \frac{d}{dx}\,\frac{1}{x} = -\frac{1}{x^2} The derivative of 1/x is -1/x². See the limit definition proof, the power rule with x⁻¹, the chain rule for 1/u and worked examples like 1/(2x + 5). Derivative of 1/x² \frac{d}{dx}\,\frac{1}{x^2} = -\frac{2}{x^3} The derivative of 1/x² is -2/x³. See the power rule and quotient rule proofs, the chain rule for 1/u², a graph and examples like 1/(3x - 2)². Derivative of |x| \frac{d}{dx}\,\lvert x\rvert = \frac{x}{\lvert x\rvert} The derivative of |x| is x/|x|: 1 for x > 0, -1 for x < 0, and undefined at 0. See the proof, the corner at 0 and the chain rule for |u|. Derivative of a constant \frac{d}{dx}\,c = 0 The derivative of a constant is 0. See why with the limit definition, plus tricky cases like π², e³, ln 5 and constants multiplied by functions.

Derivative of sin x \frac{d}{dx}\,\sin x = \cos x The derivative of sin x is cos x. See the limit proof, the chain rule version for sin(u), a graph and worked examples like sin(3x) and x sin x. Derivative of cos x \frac{d}{dx}\,\cos x = -\sin x The derivative of cos x is -sin x. See the limit proof, a quick proof from sin x, the chain rule for cos(u) and worked examples like cos(5x). Derivative of tan x \frac{d}{dx}\,\tan x = \sec^2 x The derivative of tan x is sec²x. See the quotient rule proof, the 1 + tan²x form, the chain rule for tan(u) and examples like tan(4x) and tan²x. Derivative of sec x \frac{d}{dx}\,\sec x = \sec x\tan x The derivative of sec x is sec x tan x. See the proof from 1/cos x, the chain rule for sec(u), a graph and worked examples like sec(2x) and x sec x. Derivative of csc x \frac{d}{dx}\,\csc x = -\csc x\cot x The derivative of csc x is -csc x cot x. See the proof from 1/sin x, the chain rule for csc(u), a graph and worked examples like csc(3x). Derivative of cot x \frac{d}{dx}\,\cot x = -\csc^2 x The derivative of cot x is -csc²x. See the quotient rule proof, the -(1 + cot²x) form, the chain rule for cot(u) and examples like cot(2x). Derivative of sec²x \frac{d}{dx}\,\sec^2 x = 2\sec^2 x\tan x The derivative of sec²x is 2sec²x tan x. See two proofs, one with the chain rule and one with 1 + tan²x, plus examples like sec²(3x). Derivative of sin²x \frac{d}{dx}\,\sin^2 x = 2\sin x\cos x = \sin 2x The derivative of sin²x is 2 sin x cos x, which equals sin 2x. See the chain rule proof, a check with a double angle identity and examples. Derivative of cos²x \frac{d}{dx}\,\cos^2 x = -2\sin x\cos x = -\sin 2x The derivative of cos²x is -2 sin x cos x, which equals -sin 2x. See the chain rule proof, an identity check and examples like cos²(4x). Derivative of sin 2x \frac{d}{dx}\,\sin 2x = 2\cos 2x The derivative of sin 2x is 2cos 2x. See the chain rule proof, a check with 2 sin x cos x, the general sin(ax) rule and worked examples. Derivative of sin(x²) \frac{d}{dx}\,\sin(x^2) = 2x\cos(x^2) The derivative of sin(x²) is 2x cos(x²). See the chain rule step by step, how it differs from sin²x, the second derivative and worked examples.

Derivative of arcsin x \frac{d}{dx}\,\arcsin x = \frac{1}{\sqrt{1 - x^2}} The derivative of arcsin x is 1/√(1 − x²). See the proof from sin y = x, the chain rule form for arcsin(u), a graph and worked examples. Derivative of arccos x \frac{d}{dx}\,\arccos x = -\frac{1}{\sqrt{1 - x^2}} The derivative of arccos x is −1/√(1 − x²). See the proof, why it is the negative of the arcsin derivative, chain rule examples and a graph. Derivative of arctan x \frac{d}{dx}\,\arctan x = \frac{1}{1 + x^2} The derivative of arctan x is 1/(1 + x²). See the proof from tan y = x, chain rule examples like arctan(2x) and arctan(x²), and a graph. Derivative of arcsec x \frac{d}{dx}\,\operatorname{arcsec} x = \frac{1}{\lvert x\rvert\sqrt{x^2 - 1}} The derivative of arcsec x is 1/(|x|√(x² − 1)). See where the absolute value comes from, a proof via arccos(1/x), and chain rule examples. Derivative of arccot x \frac{d}{dx}\,\operatorname{arccot} x = -\frac{1}{1 + x^2} The derivative of arccot x is −1/(1 + x²). See the proof from cot y = x, the link to arctan, chain rule examples and a graph.

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).

Derivative of sinh x \frac{d}{dx}\,\sinh x = \cosh x The derivative of sinh x is cosh x. See the proof from the exponential definition, why there is no minus sign, chain rule examples and a graph. Derivative of cosh x \frac{d}{dx}\,\cosh x = \sinh x The derivative of cosh x is sinh x, with no minus sign. See the proof from exponentials, the hanging cable example, chain rule examples and a graph. Derivative of tanh x \frac{d}{dx}\,\tanh x = \operatorname{sech}^2 x The derivative of tanh x is sech² x, which equals 1 − tanh² x. See the quotient rule proof, chain rule examples, a graph and the arctanh derivative.