AP Calculus AB and BC

Derivative of sin 4x: Answer and Chain Rule

The derivative of sin 4x is 4 cos 4x. The outer sine differentiates to cosine with the argument unchanged, then the chain rule multiplies by 4, the derivative of the inner function.

ddx[sin⁡4x]=4cos⁡4x\frac{d}{dx}\left[\sin 4x\right] = 4\cos 4x

The chain rule coefficient

ddxsin⁡(kx)=kcos⁡(kx)  ⟹  ddxsin⁡(4x)=4cos⁡4x\frac{d}{dx}\sin(kx) = k\cos(kx) \implies \frac{d}{dx}\sin(4x) = 4\cos 4x

Quadrupling the input makes the wave oscillate four times as fast, so its slopes are four times as steep. The factor is the actual rate, not bookkeeping.

Reversed, the factor divides

∫sin⁡4x dx=−cos⁡4x4+C\int \sin 4x\,dx = -\frac{\cos 4x}{4} + C

Whether the coefficient multiplies or divides is the fastest check on which direction you are working in.

Common mistakes

  • Answering cos⁡4x\cos 4x and dropping the chain rule factor.
  • Answering 4cos⁡x4\cos x, moving the coefficient out of the argument.
  • Confusing it with 4sin⁡x4\sin x, whose derivative is 4cos⁡x4\cos x.

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 sin 4x?

It is 4cos⁡4x4\cos 4x.

What is the antiderivative?

−cos⁡4x4+C-\frac{\cos 4x}{4} + C. The same 44 divides going backwards.