CAIE A-Level Mathematics 3.4.2 Product and Quotient Rules

Practise differentiating algebraic, exponential and trigonometric products or quotients, factorising the result and solving exact stationary-point or proof conditions.

Exam points

apply uv′ + vu′ or (vu′ − uv′)/v² while retaining every chain-rule factor
factor or simplify the derivative numerator before setting it equal to zero
solve the resulting algebraic or trig condition and substitute back for full coordinates

[Maximum number: 4]

Figure for Question 3.4.2—Product and quotient rules question 1 — CAIE A-Level Mathematics A2

The diagram shows the curve $y=x \mathrm{e}^{-\frac{1}{4} x^{2}}$ , for $x \geqslant 0$ , and its maximum point M.

Find the exact coordinates of M.

Use the correct product rule

*M1

Condone error in chain rule.

Obtain correct derivative in any form

A1

e.g. $\frac{d y}{d x}=-\frac{x^{2}}{2} e^{-\frac{x^{2}}{4}}+e^{-\frac{x^{2}}{4}}$ .

Equate derivative to zero and solve for x

DM1

Obtain answer $\left(\sqrt{2}, \sqrt{2} \mathrm{e}^{-\frac{1}{2}}\right)$

A1

Or exact equivalent.
Can state the components separately.

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3.4.2—Product and quotient rules question 1