Worked example 1
Differentiate a composite power
Differentiate y = (3x − 1)⁴
- Differentiate the outer fourth power.
- Keep the inner expression 3x − 1.
- Multiply by the inner derivative 3.
Answer: dy/dx = 12(3x − 1)³
Higher Mathematics
Higher Maths Chain rule
Practise chain rule for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Differentiate composite algebraic and trigonometric functions using the chain rule.
Before you start
- Differentiate powers and trig functions.
- Identify inner and outer functions.
- Keep composite expressions bracketed.
Explanation
The chain rule differentiates a function inside another function. Differentiate the outer function while keeping the inner expression, then multiply by the derivative of the inner function.
For powers of a linear expression, this creates the rule d/dx[(px + q)n] = np(px + q)n − 1.
Method and rules
- If y = f(g(x)), then dy/dx = f'(g(x))g'(x).
- d/dx[(px + q)n] = np(px + q)n − 1
- d/dx[sin(g(x))] = cos(g(x))g'(x)
Worked examples
Worked example 2
Method check
Differentiate (2x + 5)³
- Multiply by the inner derivative 2.
So: 6(2x + 5)²
Watch out
- Differentiating only the outer function.
- Expanding unnecessarily and creating algebra errors.
- Changing the inner expression while reducing the outer power.
Exam reminder
For chain rule, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
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