Mathematical Methods — Question 11

QCAA 2021, Paper 1 · 5 marks

Q11 · 2021 · Technology-freeSimple familiar5 marks

QUESTION 11 (5 marks)

Determine the derivative with respect to

x

of the following functions.

y=(e^x+1)^3

[2 marks]

y=\dfrac{\sin(x)}{x^2}

(Give your answer in simplest form.)

[3 marks]

WORKED SOLUTION

5 marks

ANSWER

3e^x(e^x+1)^2

; b)

\dfrac{x\cos x-2\sin x}{x^3}

Worked solution

a) Let

u=e^x+1

. Using the chain rule,

\frac{dy}{dx}=3e^x(e^x+1)^2.

b) Using the quotient rule,

\frac{dy}{dx}=\frac{x^2\cos x-2x\sin x}{x^4}=\frac{x\cos x-2\sin x}{x^3}.

correctly identifies the use of the chain rule

[1 mark]

correctly determines the derivative

[1 mark]

correctly identifies the use of the quotient rule

[1 mark]

correctly determines the derivative

[1 mark]

provides derivative in simplest form

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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