AP Calculus AB and BC

Derivative of x arccos x: Product Rule

The derivative of x times arccos x is arccos x minus x over the square root of 1 minus x squared. The product rule gives arccos x from differentiating the x, and the second term is x times the arccosine derivative, which carries a minus sign.

ddx[xarccos⁡x]=arccos⁡x−x1−x2\frac{d}{dx}\left[x\arccos x\right] = \arccos x - \frac{x}{\sqrt{1-x^{2}}}

Product rule with an inverse trig factor

ddx(xarccos⁡x)=arccos⁡x+x⋅(−11−x2)\frac{d}{dx}\left(x\arccos x\right) = \arccos x + x\cdot\left(-\frac{1}{\sqrt{1-x^{2}}}\right)

The minus comes from the arccosine derivative, not from the product rule, which only ever adds its two terms.

Domain and endpoints

The function is defined on −1≤x≤1-1 \le x \le 1, but the derivative fails at both endpoints where the radical vanishes. Near x=1x = 1 the slope runs to negative infinity.

Common mistakes

  • Writing a plus before the fraction, which is the arcsine version.
  • Multiplying the derivatives instead of using the product rule.

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 x arccos x?

It is arccos⁡x−x1−x2\arccos x - \frac{x}{\sqrt{1-x^{2}}}.

What is the domain?

The function needs −1≤x≤1-1 \le x \le 1; the derivative additionally excludes both endpoints.