AP Calculus AB and BC

Derivative of cos x / x: Quotient Rule

The derivative of cos x over x is negative x sin x minus cos x, all over x squared. The quotient rule gives it directly, and both terms in the numerator end up negative because cosine differentiates to negative sine.

ddx[cos⁡xx]=−xsin⁡x−cos⁡xx2\frac{d}{dx}\left[\frac{\cos x}{x}\right] = \frac{-x\sin x - \cos x}{x^{2}}

Applying the quotient rule

(−sin⁡x)x−cos⁡x⋅1x2=−xsin⁡x−cos⁡xx2\frac{\left(-\sin x\right)x - \cos x\cdot 1}{x^{2}} = \frac{-x\sin x - \cos x}{x^{2}}

Factoring a minus gives −xsin⁡x+cos⁡xx2-\frac{x\sin x + \cos x}{x^{2}}, which some answer keys prefer. The two are identical.

A bounded numerator over a growing denominator

As x→∞x \to \infty the function tends to 00, because cosine stays bounded while the denominator grows. The oscillations shrink inside a ±1x\pm\frac{1}{x} envelope.

Common mistakes

  • Forgetting that cosine differentiates to NEGATIVE sine.
  • Subtracting in the wrong order in the numerator.

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 cos x / x?

It is −xsin⁡x−cos⁡xx2\frac{-x\sin x - \cos x}{x^{2}}.

What happens at infinity?

The function tends to 00: a bounded numerator over an unbounded denominator.