Year 12 Calculus: The chain rule

The chain rule is used when one function is nested inside another. You differentiate the outer function, keep the inner part unchanged, and then multiply by the derivative of the inner function.

Think "outside first, inside second". If you can name the inner function clearly, the derivative is usually much easier to organize.

Main rule

If $y = f (g (x))$ , then $d y / d x = f' (g (x)) g' (x)$ .

Worked example

Problem: Differentiate

y = {(3 x - 1)}^{5}

Outer function: raise something to the power 5.
Inner function: $u = 3 x - 1$ .
Differentiate the outer function: $d (u^{5}) / d x = 5 u^{4}$ .
Differentiate the inner function: $d u / d x = 3$ .
Multiply them: $d y / d x = 5 {(3 x - 1)}^{4} * 3$ .
Simplify: $d y / d x = 15 {(3 x - 1)}^{4}$ .

Answer:

d y / d x = 15 {(3 x - 1)}^{4}

How to spot it quickly

The chain rule appears whenever one full expression is sitting inside another, especially inside a power, a logarithm, a trigonometric function, or an exponential. If you can point to an inner piece that could be replaced by one temporary letter, then the chain rule is probably involved.

Worked example 2

Problem: Differentiate

y = {(x^{2} + 4 x)}^{3}

Outer function: cube something.
Inner function: $u = x^{2} + 4 x$ .
Differentiate the outer function: $d (u^{3}) / d x = 3 u^{2}$ .
Differentiate the inner function: $d u / d x = 2 x + 4$ .
Multiply: $d y / d x = 3 {(x^{2} + 4 x)}^{2} (2 x + 4)$ .

Answer:

d y / d x = 3 {(x^{2} + 4 x)}^{2} (2 x + 4)

Common mistakes

Differentiating the outer function but forgetting the derivative of the inner part.
Expanding a complicated power first when the chain rule would be faster.
Misidentifying the inner function when several brackets appear.

The chain rule is important because nested structure appears everywhere in advanced calculus. Powers of brackets, logarithms of expressions, trigonometric functions of linear terms, and exponentials of polynomials all depend on the same basic idea. If you can spot the inner function quickly, you reduce a complicated derivative to a short repeated pattern instead of treating every question as new.

A strong self-check is to circle the inner function before you start. Then, after differentiating, ask whether the derivative contains two parts: the derivative of the outer layer and the derivative of that inner layer. If only one of those appears, the chain rule is incomplete.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.

It is also worth building a habit of interpretation after the algebra is finished. A derivative is not just a formula. It represents a slope, a rate, or a turning condition. An integral is not just a reverse power rule. It may represent accumulated change or a signed area over an interval. If you say that interpretation out loud at the end of each question, the topic becomes much more coherent and your solutions become much easier to check.