Differentiate composite functions with the chain rule
Differentiate a function of a linear or inner expression by multiplying by the derivative of that inner expression.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In the chain rule for F(g(x)), what extra factor appears?
What you will learn
- Use the chain rule for composite functions
- Differentiate powers and standard functions of ax + b
- Keep radians in composite trigonometric functions
Differentiate the outside and multiply by the inside rate
If y = F(g(x)), then dy/dx = F′(g(x))g′(x). First differentiate the outer function while leaving the inner expression in place; then multiply by the derivative of the inner expression.
For y = (ax + b)ⁿ, dy/dx = an(ax + b)ⁿ⁻¹. For sin(ax + b), the derivative is a cos(ax + b); the factor a comes from differentiating the inside.
The same rule handles exponential, logarithmic and trigonometric forms
d(e^(ax+b))/dx = a e^(ax+b), d(ln(ax+b))/dx = a/(ax+b), and d(tan(ax+b))/dx = a sec²(ax+b), wherever each original function is defined.
For sin and cos of a composite angle, use radians and multiply by the inner derivative. Do not leave out a constant factor just because the inner expression is in brackets.
Pause and explain
Differentiate (2x + 1)³.
Worked example
Differentiate y = (3x − 2)⁵.
Show the worked solution
- Differentiate the outer fifth power to get 5(3x − 2)⁴.
- The inner derivative of 3x − 2 is 3.
- Multiply by the inner derivative.
Answer dy/dx = 15(3x − 2)⁴.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Differentiate y = √(5x + 1), writing the answer with a positive power if possible.
Give me a hint
Write the square root as a power and apply the chain rule.
Compare my reasoning
- y = (5x + 1)^(1/2).
- dy/dx = (1/2)(5x + 1)^(−1/2) × 5.
- Simplify the constant factor.
dy/dx = 5/[2√(5x + 1)].
Look for these in your work
- Power rewritten
- Inner derivative included
- Equivalent form simplified
Differentiate y = e^(3x−2) + ln(2x + 5).
Give me a hint
Apply the inner derivative to both terms and retain the logarithm's domain.
Compare my reasoning
- d(e^(3x−2))/dx = 3e^(3x−2).
- d(ln(2x+5))/dx = 2/(2x+5).
- The logarithm requires 2x + 5 > 0.
dy/dx = 3e^(3x−2) + 2/(2x + 5), for x > −5/2.
Look for these in your work
- Exponential chain factor included
- Logarithm chain factor included
- Domain condition stated
Differentiate y = sin(2x + π/3) for x measured in radians.
Give me a hint
The outer derivative is cosine; multiply by the derivative of the angle.
Compare my reasoning
- The derivative of sin(u) is cos(u) du/dx.
- Here u = 2x + π/3 and du/dx = 2.
- Multiply by 2 and retain the original form.
dy/dx = 2 cos(2x + π/3).
Look for these in your work
- Radians used
- Inner angle differentiated
- Chain factor included
Common mistakes
- Forgetting the inner derivative
- Expanding a large power before differentiating unnecessarily
- Changing the inner expression while applying the outer rule
- Using degree-mode trig derivatives
What can you explain now?
What is the derivative of sin(4x)?
Compare with the explanation
4 cos(4x)
Differentiate sine, then multiply by d(4x)/dx = 4.
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Make it stick
What are the two steps in the chain rule, and where does the inner derivative appear?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.