AP Calculus AB and BC

Derivative of arccos x: Answer, Proof, and Mistakes

The derivative of arccos x with respect to x is negative one over the square root of one minus x squared. It is the exact negative of the derivative of arcsin x, because the two functions sum to a constant, and a constant has zero derivative.

ddx[arccos⁡x]=−11−x2\frac{d}{dx}\left[\arccos x\right] = -\frac{1}{\sqrt{1 - x^{2}}}

How to differentiate arccos x

The fastest derivation uses a identity rather than implicit differentiation. Arcsine and arccosine are complementary.

arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2}

Differentiating both sides gives zero on the right, so the two derivatives must be negatives of each other.

ddxarccos⁡x=−11−x2\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1 - x^{2}}}

What the formula tells you about the graph

The derivative is negative on the whole domain −1<x<1-1 < x < 1, so arccosine is strictly decreasing, running from π\pi down to 00.

As xx approaches ±1\pm 1 the denominator approaches zero and the slope becomes infinitely steep, which is a vertical tangent at each endpoint.

Where the derivative of arccos x shows up on the AP exam

Topic 3.4 (Differentiating Inverse Trigonometric Functions) on both exams. It is worth knowing that arccosine is deliberately NOT the standard antiderivative of anything on the formula sheet, because arcsine already covers that integrand.

Common mistakes with the derivative of arccos x

  • Dropping the minus sign, which gives the arcsine derivative instead.
  • Writing −1x2−1\frac{-1}{\sqrt{x^{2}-1}}. The order under the radical matters: it is 1−x21 - x^{2}, which is positive on the domain.
  • Confusing it with ddx(cos⁡x)−1\frac{d}{dx}\left(\cos x\right)^{-1}, the reciprocal, which is a different function entirely.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of arccos x?

It is −11−x2-\frac{1}{\sqrt{1 - x^{2}}}.

Why is it the negative of the arcsin derivative?

Because arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2} is constant, so differentiating gives 00 and the two derivatives must cancel.

What is the domain of the derivative?

Strictly between −1-1 and 11. At the endpoints the denominator is zero and the tangent line is vertical.

Is arccos x increasing or decreasing?

Decreasing everywhere on its domain, which is exactly what the negative derivative says.