Q.Differentiate $y=(2x^3-5x+1)^6$ 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: …