AP Calculus AB and BC

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

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

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

Q: Is sin(x^2) the same as sin^2 x?

No. \sin(x^{2}) differentiates to 2x\cos(x^{2}), while \sin^{2}x = (\sin x)^{2} differentiates to 2\sin x\cos x.

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

-2x\sin(x^{2}), the same chain rule with the outer sine replaced by cosine, which brings a minus sign.

The derivative of sin of x squared with respect to x is 2x cos(x^2). The inner function is x squared with derivative 2x, and the outer function is sine, so the chain rule gives cosine of x squared times 2x. This is not the same as (sin x) squared, whose derivative is 2 sin x cos x.

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

How to differentiate sin(x^2)

In $\sin(x^{2})$ the squaring happens first, inside the sine, so $x^{2}$ is the inner function and sine is the outer function. The chain rule differentiates the outer, keeps the inner, then multiplies by the inner derivative.

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

With $u = x^{2}$ , the inside differentiates to $2x$ .

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

Not the same as sin^2 x

Read the notation carefully

$\sin(x^{2})$ takes the sine of $x^{2}$ and differentiates to $2x\cos(x^{2})$ . The look-alike $\sin^{2}x$ means $(\sin x)^{2}$ and differentiates to $2\sin x\cos x$ . The square sits in different places, and the two answers are different functions.

A quick numeric check separates them: at $x = 1$ , the derivative $2x\cos(x^{2})$ equals $2\cos 1 \approx 1.08$ , while $2\sin x\cos x$ equals $\sin 2 \approx 0.91$ . Different values confirm they are not the same rule.

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

The chain rule is Topic 3.1, on both AB and BC. Telling $\sin(x^{2})$ apart from $\sin^{2}x$ is a standard multiple choice trap, since the notation differs only by where the exponent sits.

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

Answering $\cos(x^{2})$ , forgetting the inner derivative $2x$ .
Answering $2\sin x\cos x$ , reading $\sin(x^{2})$ as $\sin^{2}x$ .
Answering $2x\cos x$ , applying the chain factor but leaving the argument as $x$ instead of $x^{2}$ .

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(x^2)?

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

Is sin(x^2) the same as sin^2 x?

No. $\sin(x^{2})$ differentiates to $2x\cos(x^{2})$ , while $\sin^{2}x = (\sin x)^{2}$ differentiates to $2\sin x\cos x$ .

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

$-2x\sin(x^{2})$ , the same chain rule with the outer sine replaced by cosine, which brings a minus sign.