Apply chain rule to differentiate composite function sin(x²).

Find dy/dx of sin(x²) — Chain Rule

Question

Find $\dfrac{dy}{dx}$ if $y = \sin(x^2)$ .

This is a straightforward chain rule question that appeared in CBSE 2024 Board Exam and is a favourite warm-up in JEE Main too.

Solution — Step by Step

We have an outer function and an inner function packed together. Write it out clearly:

Outer function: $\sin(\square)$
Inner function: $x^2$

Recognising this structure before differentiating is half the work done.

For $y = f(g(x))$ , the chain rule says:

\frac{dy}{dx} = f'(g(x)) \cdot g'(x)

Plain English: differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.

The outer function is $\sin(\square)$ . Its derivative is $\cos(\square)$ .

So differentiating $\sin(x^2)$ with respect to the inner function gives us $\cos(x^2)$ .

The inner function $x^2$ stays as-is inside the cosine — we haven’t touched it yet.

Now differentiate the inner function $x^2$ :

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

Multiply this onto what we got in Step 3:

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

Rearranging (convention puts the coefficient first):

\boxed{\frac{dy}{dx} = 2x\cos(x^2)}

Why This Works

The chain rule exists because $x^2$ is itself changing as $x$ changes. If we just wrote $\cos(x^2)$ and stopped, we’d be treating $x^2$ as a constant — which it isn’t. The factor $2x$ accounts for how fast the input to the sine function is changing.

Think of it this way: the rate at which the whole expression $\sin(x^2)$ changes depends on two things — how fast $\sin$ responds to its input, and how fast that input ( $x^2$ ) is itself moving. The chain rule multiplies these two rates together.

This is why the chain rule is non-negotiable for any composite function. In CBSE and JEE, roughly 30–40% of differentiation problems require it in some form.

Alternative Method — Substitution

Some students find it cleaner to introduce a substitution explicitly:

Let $t = x^2$ , so $y = \sin(t)$ .

Now differentiate using the chain rule in Leibniz form:

\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}

We know $\dfrac{dy}{dt} = \cos(t)$ and $\dfrac{dt}{dx} = 2x$ .

Substituting back ( $t = x^2$ ):

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

Same answer, different notation. The substitution method is slower but less error-prone when the composite structure is deeply nested — useful for functions like $\sin(\sqrt{x^2 + 1})$ .

Common Mistake

The single most common error here is writing $\dfrac{dy}{dx} = \cos(2x)$ — differentiating inside the argument while applying the outer derivative.

What went wrong: the student differentiated $x^2 \to 2x$ correctly, but then replaced $x^2$ with $2x$ inside the cosine. The inner function must stay unchanged when you apply the outer derivative. You differentiate the outer, then multiply — never substitute inside.

Correct: $\cos(x^2) \cdot 2x$ . Wrong: $\cos(2x)$ .

Quick self-check: count the number of functions composed together. For $\sin(x^2)$ — that’s 2 functions, so you need exactly one chain rule factor ( $2x$ ). For $\sin(\cos(x^2))$ — that’s 3 functions, so you need two chain rule factors. If your derivative has fewer multiplicative factors than expected, you’ve missed a chain.

Question

Solution — Step by Step

Why This Works

Alternative Method — Substitution

Common Mistake

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