Mathematical Methods — Question 15

QCAA 2023, Paper 2 · 4 marks

Q15 · 2023 · Technology-activeSimple familiar4 marks

QUESTION 15 (4 marks)

Determine the derivative of

f(x)=\ln x^2+\ln(x-5)^3

. Express the derivative as a single fraction in its simplest and factorised form.

WORKED SOLUTION

4 marks

ANSWER

f'(x)=\dfrac{5(x-2)}{x(x-5)}

Worked solution

f(x)=\ln x^2+\ln(x-5)^3=2\ln x+3\ln(x-5).

Therefore

f'(x)=\frac2x+\frac3{x-5}.

Combining the fractions,

f'(x)=\frac{2(x-5)+3x}{x(x-5)}=\frac{5x-10}{x(x-5)}=\frac{5(x-2)}{x(x-5)}.

correctly determines first term derivative

[1 mark]

correctly determines second term derivative

[1 mark]

combines the fractions to one fraction

[1 mark]

determines the simplest and factorised form

[1 mark]

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

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