Cambridge O Level Additional Mathematics · Syllabus 4037 · Calculus
Chain Rule
What is Chain Rule?
The rule for differentiating a composite function, that is, a function of a function. If y equals f of g of x, then dy by dx equals f dashed of g of x multiplied by g dashed of x. In words: differentiate the outer function while leaving the inner function untouched, then multiply by the derivative of the inner function. The extra factor is needed because a change in x first changes the inner function and that change then changes y, so the two rates multiply. In Leibniz notation the same rule reads dy by dx equals dy by du multiplied by du by dx. It is not supplied in the examination formula list.
This definition is part of the Calculus chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Chain Rule
Why do you need the chain rule to differentiate \((3x^2+1)^4\)?
Because the power rule alone differentiates a power of \(x\), and here the quantity being raised to a power is itself a function of \(x\), a bracket that changes as \(x\) changes. The chain rule accounts for that inner change: differentiate the outer power to get \(4(3x^2+1)^3\), then multiply by the derivative of the inner function, \(6x\), giving \(24x(3x^2+1)^3\). Forgetting that extra factor is the single most common differentiation error in this chapter.