Q.Differentiate $y=\cos(5x-3)$ with respect to $x$.

The Chain Rule: Why It Makes Sense

Imagine you're assembling a toy. First you put part A into part B, then you put that combined piece into part C. The final toy's position depends on how you moved A, which then affected B, which then affected C. That's exactly what the chain rule captures — how a change in the first variable ripples through a sequence of functions to affect the final output.

Let's make this concrete. Suppose you have a function $f$ that depends on $g$, and $g$ itself depends on $x$:

You want to know: if $x$ changes by a tiny amount, how much does $y$ change? The answer isn't just $f'(g(x))$ — because $g(x)$ itself changes when $x$ changes. You have to multiply the two rates:

• How fast does $g$ change with respect to $x$? That's $g'(x)$.
• How fast does $f$ change with respect to its input $g$? That's $f'(g(x))$.

The total effect is the product:

The Precise Statement

Chain Rule (single variable): If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then the composite function $h(x) = f(g(x))$ is differentiable at $x$, and

That's it. One multiplication. But the power is enormous — it lets you differentiate almost any nested function.

A Simple Example

Differentiate $h(x) = \sin(3x^2)$.

Here $f(u) = \sin u$ and $g(x) = 3x^2$. Then:

• $f'(u) = \cos u$, so $f'(g(x)) = \cos(3x^2)$
• $g'(x) = 6x$

Multiply: $h'(x) = \cos(3x^2) \cdot 6x = 6x \cos(3x^2)$

Why It's Called a "Chain"

Think of a chain of links: $x \to g \to f$. Each link has its own rate of change. To find the total rate from $x$ to $f$, you multiply the rates of each link. If you had three functions — say $h(x) = f(g(k(x)))$ — you'd multiply three derivatives:

The chain can be as long as you like. Each new function adds one more factor.

The Intuition in One Sentence

The chain rule says: the rate of change of the whole is the product of the rates of change of the parts, evaluated at the right places.

The chain rule is one of the most heavily tested formulas in the NCERT Class 12 Continuity and Differentiability chapter, and it underlies nearly every differentiation problem in CBSE boards, JEE Main and JEE Advanced. Whether you're searching 'chain rule differentiation class 12 examples' or 'chain rule important questions for JEE', this f'(g(x))·g'(x) pattern is the formula every subsequent derivative rule in the syllabus builds on.