AP Calculus AB and BC

Derivative of cos(x^2): Answer, Chain Rule, Mistakes

Q: What is the derivative of cos(x^2)?

\frac{d}{dx}\cos(x^{2}) = -2x\sin(x^{2}), by the chain rule with inner function x^{2}.

Q: Why is there a minus sign?

Differentiating the outer cosine gives -\sin, and the chain rule attaches the inner derivative 2x, producing -2x\sin(x^{2}).

Q: How is cos(x^2) different from cos^2 x?

\cos(x^{2}) squares the input and gives -2x\sin(x^{2}); \cos^{2}x squares the output and gives -2\cos x\sin x.

The derivative of cos of x squared with respect to x is negative 2x sin(x^2). The inner function is x squared with derivative 2x, and differentiating the outer cosine brings a minus sign, so the chain rule gives -2x sin(x^2). This differs from (cos x) squared, whose derivative is -2 cos x sin x.

\frac{d}{dx}\left[\cos(x^2)\right] = -2x\sin(x^2)

How to differentiate cos(x^2)

The squaring is inside the cosine, so $x^{2}$ is the inner function and cosine is the outer function. The chain rule differentiates the outer, keeps the inner, then multiplies by the inner derivative.

\frac{d}{dx}\cos(u) = -\sin(u)\cdot\frac{du}{dx}

With $u = x^{2}$ , the inside differentiates to $2x$ , and the outer cosine contributes a minus sign.

\frac{d}{dx}\cos(x^{2}) = -\sin(x^{2})\cdot 2x = -2x\sin(x^{2})

Two things happen at once

Track the minus sign and the 2x

The minus sign comes from differentiating cosine, and the factor $2x$ comes from the chain rule acting on the inner $x^{2}$ . Dropping either one is the usual error.

Do not read $\cos(x^{2})$ as $\cos^{2}x = (\cos x)^{2}$ . That look-alike differentiates to $-2\cos x\sin x$ , a different function. In $\cos(x^{2})$ the square is on the input; in $\cos^{2}x$ it is on the output.

Where the derivative of cos(x^2) shows up on the AP exam

The chain rule is Topic 3.1, on both AB and BC. Like its sine partner, $\cos(x^{2})$ is used to test whether you can identify the inner function and keep the argument intact through differentiation.

Common mistakes with the derivative of cos(x^2)

Answering $-\sin(x^{2})$ , forgetting the inner derivative $2x$ .
Answering $2x\sin(x^{2})$ , losing the minus sign that comes from the cosine.
Answering $-2\cos x\sin x$ , reading $\cos(x^{2})$ as $\cos^{2}x$ .

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 cos(x^2)?

$\frac{d}{dx}\cos(x^{2}) = -2x\sin(x^{2})$ , by the chain rule with inner function $x^{2}$ .

Why is there a minus sign?

Differentiating the outer cosine gives $-\sin$ , and the chain rule attaches the inner derivative $2x$ , producing $-2x\sin(x^{2})$ .

How is cos(x^2) different from cos^2 x?

$\cos(x^{2})$ squares the input and gives $-2x\sin(x^{2})$ ; $\cos^{2}x$ squares the output and gives $-2\cos x\sin x$ .