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Trigonometric derivatives

Derivative of tan(x): Formula, Proof, and Examples

The tangent derivative is easiest to verify by writing tan(x)=sin(x)/cos(x). The same points excluded from tangent remain excluded from its derivative formula.

Direct answer

ddx[tan⁡x]=sec⁡2x\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan x\right] = \sec^{2} x

Requires cos(x) ≠ 0; angles are measured in radians.

Proof with the quotient rule

Rewrite tangent as sin(x)/cos(x), then differentiate the quotient.

The numerator becomes cos²(x)+sin²(x), which equals 1. The denominator is cos²(x).

Thus the derivative is 1/cos²(x)=sec²(x), at points where cos(x) is nonzero.

1
(sin x / cos x)' = (cos²x + sin²x)/cos²x
2
= 1/cos²(x)
3
= sec²(x)

Worked examples

ddx[tan⁡(3 x)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan\left(3\,x\right)\right]
3 sec2 3 x3\,\mathrm{sec}^{2}\,3\,x

Multiply by the inner derivative 3.

ddx[tan⁡(x2)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan\left(x^{2}\right)\right]
2 x sec⁡2(x2)2\,x\,\sec^{2}\left(x^{2}\right)

Keep the whole inner input inside sec².

ddx[tan⁡2x]\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan^{2} x\right]
2 tan⁡x sec⁡2x2\,\tan x\,\sec^{2} x

Use the power rule followed by the tangent rule.

Common mistakes

  • • Writing sec(x) instead of sec²(x).
  • • Omitting the restriction cos(x) ≠ 0.
  • • Forgetting the inner derivative in tan(g(x)).

Try the worked examples in the calculator

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Derivadas:tan⁡(3 x)\tan\left(3\,x\right)
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