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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).

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Student-friendly explanation

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

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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).

Example

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

Rule to remember

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.

Memory hook

Differentiate outside, keep inside, then multiply by inside derivative.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

Differentiate y = (3x^2 - 5)^4 with respect to x.

dy/dx = 4(3x^2 - 5)^3 * 6x = 24x(3x^2 - 5)^3.

Answer writing and exam use

1-mark answer

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-mark answer

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

3-mark answer

Use the chain rule when one function is placed inside another, such as sin(x^2), e^(3x+1), log(1+x^2), or (2x+5)^7. The inner function must also be differentiated; missing this multiplier is the most common loss of marks. 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. Differentiate y = cos(5x^3 - 2x). Let u = 5x^3 - 2x. Then y = cos u. dy/du = -sin u and du/dx = 15x^2 - 2. Therefore dy/dx = -sin(5x^3 - 2x)(15x^2 - 2). Appears as direct differentiation, part of implicit differentiation, or combined with inverse trigonometric, logarithmic, and exponential functions. Writing d/dx[(2x+1)^5] = 5(2x+1)^4 instead of 10(2x+1)^4.
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