Choosing a Differentiation Rule | CAIE 9709 Pure Mathematics 2 Notes

Chapters

Pure Mathematics 2

Mathematics9709

Read the outer operation

What is the outer structure?

Product

two changing factors

u v′ + v u′

Quotient

function over function

(v u′ − u v′) / v²

Composite

function inside function

outside′ × inside′

Simplify first if it reveals an easier structure.

Choose the rule from the structure before doing any algebra.

A sum or difference is differentiated term by term. The plus or minus signs stay between the derivative terms.
$5f(x)$ is a constant multiple. Keep 5; it is not a second changing factor.
$(3x-1)^5$ is a composite: one function is inside another. It needs the chain rule.
$x^2\ln x$ is a product because both written factors change with $x$ .
$\sin x/(x+2)$ is a quotient because one changing function is divided by another.

Outer structure

simplify

Cancel x − 1 first, then differentiate x + 1.

The common factor removes the quotient for x ≠ 1.

Select an expression. Read its outer structure before choosing a rule.

Simplify first

Algebra may expose an easier form. For example,

\frac{x^2-1}{x-1}=x+1\qquad(x\ne1)

The derivative is 1 wherever the original function is defined. The condition $x\ne1$ remains: cancellation does not fill the hole in the original graph.

Your turnIndependent check [5 marks]

State the first rule or action you would use for each expression: $4e^x$ , $(2x+1)^6$ , $x\cos x$ , $\ln x/(x+1)$ and $(x^2-4)/(x-2)$ .

Show worked answer

$4e^x$ : constant multiple.
$(2x+1)^6$ : chain rule.
$x\cos x$ : product rule.
$\ln x/(x+1)$ : quotient rule.
$(x^2-4)/(x-2)$ : simplify first.