medium difficulty7 min

Chain Rule for Composite Functions

The chain rule gives the derivative of a composite function f(g(x)) by differentiating the outer function at g(x) and multiplying by the derivative of g(x).

In one line: Differentiate outside, keep inside, then multiply by inside derivative.

Practice This Concept

Understand it

Use the chain rule when one function is placed inside another, such as sin(x²), e^3x+1, log(1+x²), or (2x+5)⁷. The inner function must also be differentiated; missing this multiplier is the most common loss of marks.

Example

If y = sin(x² + 1), then dy/dx = cos(x² + 1) × 2x.

Formula and rule

If y = f(g(x)), then dy/dx = f'(g(x))g'(x).

Conditions: f must be differentiable at g(x), and g must be differentiable at x.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Differentiate y = cos(5x³ - 2x).

1Let u = 5x³ - 2x.
2Then y = cos u. dy/du = -sin u and du/dx = 15x² - 2.
3Therefore dy/dx = -sin(5x³ - 2x)(15x² - 2).

Method you can reuse

1Identify the outer function and inner function.
2Differentiate the outer function while keeping the inner expression unchanged.
3Multiply by the derivative of the inner expression.
4Simplify only after all derivative factors are included.

Visual tip

No diagram is needed; the structure is identified by nested functions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Differentiate y = (3x² - 5)⁴ with respect to x.

Think first, then show answer

dy/dx = 4(3x² - 5)³ × 6x = 24x(3x² - 5)³.

Common wrong answer

Writing d/dx[(2x+1)⁵] = 5(2x+1)⁴ instead of 10(2x+1)⁴.

Don't mix this up

Students differentiate the outer function correctly but forget to multiply by the derivative of the inner function.

Exam tip

Circle the inner function mentally before differentiating; the final derivative must contain the derivative of that inner part.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Appears as direct differentiation, part of implicit differentiation, or combined with inverse trigonometric, logarithmic, and exponential functions.

1
Enough for 1 mark
The chain rule gives the derivative of a composite function f(g(x)) by differentiating the outer function at g(x) and multiplying by the derivative of g(x).
2
Add for 2 marks
Example: If y = sin(x² + 1), then dy/dx = cos(x² + 1) × 2x.
3
Add for 3 marks
Use the chain rule when one function is placed inside another, such as sin(x²), e^3x+1, log(1+x²), or (2x+5)⁷.

MCQ Quiz

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