Q.Find the derivative of the function given by $f(x) = \sin(x^2)$.

The key idea is the chain rule: differentiate the outer function (sine) and multiply by the derivative of the inner function ($x^2$).

Step 1: Identify the outer and inner functions.

Outer: $\sin(u)$, where $u = x^2$.

Step 2: Differentiate the outer function with respect to $u$:

$\frac{d}{du} \sin(u) = \cos(u)$. …

The Chain Rule is the natural tool here because $f(x) = \sin(x^2)$ is a composition of two functions. You have an outer function, $\sin(u)$, and an inner function, $u = x^2$. The derivative of a composition is: derivative of the outer function (evaluated at the inner function) times the derivative of the inner function. In Leibniz notation, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Why does this work? Because the rate of change of $f$ with respect to $x$ depends on how fast $\sin(u)$ changes with $u$, and how fast $u$ itself changes with $x$. You multiply these rates.

Let’s apply it step by step.

1. Identify the outer and inner functions.

   Outer: $\sin(u)$, where $u$ is a placeholder.

   Inner: $u = x^2$.

2. Differentiate the outer function with respect to its argument.

   The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$.

   So $\frac{d}{du} \sin(u) = \cos(u)$.

3. Differentiate the inner function with respect to $x$.

   The derivative of $x^2$ is $2x$.

   So $\frac{du}{dx} = 2x$.

4. Apply the Chain Rule.

   Multiply the two derivatives, but remember: the outer derivative must be evaluated at the original inner function $u = x^2$, not at $u$ alone. …

Method: Differentiating a Composite Function via the Chain Rule

This method applies whenever the function to differentiate is "a function of a function" — one expression sitting inside another, such as $\sin(\text{something})$, $e^{\text{something}}$, or $(\text{something})^n$.

Steps

Step 1: Identify the outer function and the inner function

Write the given function in the form $y = f(u)$ where $u = g(x)$ is the inner expression. Ask: "if I peel away the very last operation applied, what's left inside?" That last operation is the outer function; everything it acts on is the inner function.

Step 2: Differentiate the outer function, keeping the inner function intact

Find $f'(u)$ using the standard derivative formula for the outer function's type (trig, exponential, power, etc.), but do NOT substitute or simplify the inner expression yet — just carry it along as $u$.

Step 3: Differentiate the inner function separately …

Common Mistakes

Mistake 1: Forgetting to multiply by the inner function's derivative

Why it's wrong: The chain rule requires two factors — the outer derivative evaluated at the inner function, AND the inner function's own derivative. Writing $\frac{d}{dx}\sin(x^2) = \cos(x^2)$ and stopping there drops the second factor entirely. Correct approach: always multiply by $\frac{d}{dx}(x^2) = 2x$, giving $2x\cos(x^2)$.

Mistake 2: Confusing $\sin(x^2)$ with $\sin(2x)$ …