Q.Differentiate w.r.t. $x$: $\sin x^2 + \sin^2 x + \sin^2(x^2)$.

Concept: Chain Rule — each term is a composition of $\sin$ with a power function.

Step 1: For $\sin x^2$, let $u = x^2$, then derivative is $\cos(x^2) \cdot 2x = 2x \cos x^2$.

Step 2: For $\sin^2 x$, write as $(\sin x)^2$. Derivative is $2 \sin x \cdot \cos x = \sin 2x$. …

The key to this problem is recognising that each term is a different kind of composition. You have three functions, each requiring the chain rule in a slightly different way. Let’s break them down one by one.

1. First term: $\sin x^2$

   This is $\sin$ of $(x^2)$. The outer function is $\sin(\cdot)$, the inner function is $x^2$.

   Derivative: $\cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x = 2x \cos x^2$.

2. Second term: $\sin^2 x$

   This is $(\sin x)^2$. The outer function is $(\cdot)^2$, the inner function is $\sin x$.

   Derivative: $2(\sin x) \cdot \frac{d}{dx}(\sin x) = 2\sin x \cdot \cos x = \sin 2x$ (using the double-angle identity $2\sin x \cos x = \sin 2x$).

3. Third term: $\sin^2(x^2)$

   This is $[\sin(x^2)]^2$. There are two layers of composition: first square, then sine, then $x^2$.

   • Outer: $(\cdot)^2$, derivative $2 \cdot \sin(x^2)$.
   • Middle: $\sin(x^2)$, derivative $\cos(x^2)$.
   • Inner: $x^2$, derivative $2x$. Multiply: $2 \sin(x^2) \cdot \cos(x^2) \cdot 2x = 4x \sin(x^2) \cos(x^2)$. Simplify using $2\sin\theta\cos\theta = \sin 2\theta$: $4x \cdot \frac{1}{2} \sin(2x^2) = 2x \sin(2x^2)$. …

Method: Chain Rule for Structurally Different Terms That Look Similar

Use this method when a sum contains several terms that use the same functions (sine and squaring) but combine them in different orders — such as $\sin x^2$, $\sin^2 x$, and $\sin^2(x^2)$ — so each term needs its own careful reading before differentiating.

Steps

Step 1: Rewrite each term unambiguously before differentiating anything

Translate the notation precisely: $\sin x^2$ means $\sin(x^2)$ (sine of a squared argument); $\sin^2 x$ means $(\sin x)^2$ (squared sine); $\sin^2(x^2)$ means $[\sin(x^2)]^2$ (squared sine of a squared argument). Getting this reading right is the entire difficulty of the problem.

Step 2: Differentiate the term that is a sine of a polynomial, with one chain-rule step

Step 3: Differentiate the term that is a square of a sine, with one chain-rule step

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

Common Mistakes

Mistake 1: Confusing $\sin x^2$ with $\sin^2 x$

Why it's wrong: these are genuinely different functions — $\sin x^2=\sin(x^2)$ has the squaring inside the sine, while $\sin^2 x=(\sin x)^2$ has the squaring outside — and they have different derivatives ($2x\cos x^2$ versus $\sin 2x$). Misreading one notation as the other is one of the most common exam-hall errors in this topic. Correct approach: before differentiating, explicitly rewrite each term as either sine-of-something-squared or square-of-the-sine, matching the position of the exponent $2$ exactly as printed.

Mistake 2: Applying only one chain-rule layer to $\sin^2(x^2)$ …