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2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Syllabus
2020
Topic
2.10
Level
—

Derivatives of $\tan x$, $\cot x$, $\sec x$, and $\csc x$

Rewrite tangent, cotangent, secant, and cosecant in terms of sine and cosine; then the quotient rule and known sine/cosine derivatives produce their formulas. Angles are measured in radians.

Function identity Derivative
tan⁡x=sin⁡xcos⁡x\tan x=\dfrac{\sin x}{\cos x} sec⁡2x\sec^2 x
cot⁡x=cos⁡xsin⁡x\cot x=\dfrac{\cos x}{\sin x} −csc⁡2x-\csc^2 x
sec⁡x=1cos⁡x\sec x=\dfrac{1}{\cos x} sec⁡xtan⁡x\sec x\tan x
csc⁡x=1sin⁡x\csc x=\dfrac{1}{\sin x} −csc⁡xcot⁡x-\csc x\cot x

For tangent, the quotient rule gives ddx(tan⁡x)=(cos⁡x)(cos⁡x)−(sin⁡x)(−sin⁡x)cos⁡2x=cos⁡2x+sin⁡2xcos⁡2x=sec⁡2x.\frac{d}{dx}(\tan x)=\frac{(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos^2 x}=\frac{\cos^2 x+\sin^2 x}{\cos^2 x}=\sec^2 x. The Pythagorean identity turns the numerator into 11.

The tangent/secant pair has positive derivatives. The cotangent/cosecant pair has negative derivatives. Identities explain these results; they are not four unrelated facts.

Apply each formula only where the original function is defined: tan⁡x\tan x and sec⁡x\sec x require cos⁡x≠0\cos x\ne0, while cot⁡x\cot x and csc⁡x\csc x require sin⁡x≠0\sin x\ne0. Composite inputs require the chain rule, which belongs to the next Topic.