Trigonometric derivatives

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.