CAIE A-Level Mathematics A2 3.4.2 Product and Quotient Rules Questions

Practise differentiating products and quotients of functions in the AS mathematics syllabus.

Exam points

apply the product rule to differentiate composite products
apply the quotient rule to differentiate rational functions
simplify and evaluate derivatives at specified points

[Maximum number: 4]

Figure for Question CAIE A-Level Mathematics A2 3.4.2 Product and Quotient Rules Questions 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
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
Obtain answer $\left(\sqrt{2}, \sqrt{2} \mathrm{e}^{-\frac{1}{2}}\right)$
Or exact equivalent.
Can state the components separately.

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CAIE A-Level Mathematics A2 3.4.2 Product and Quotient Rules Questions question 1