C3 June 2011 Q1

1. Differentiate with respect to \(x\)

(a) \(\ln\left(x^2 + 3x + 5\right)\) (2)

(b) \(\dfrac{\cos x}{x^2}\) (3)

Scheme	Marks
\(\dfrac{1}{(x^2+3x+5)} \times \ldots,\ = \dfrac{2x+3}{(x^2+3x+5)}\)	M1,A1
	(2)

Mark scheme (b)

Scheme	Marks
Applying \(\dfrac{vu^{\prime} - uv^{\prime}}{v^2}\)	M1,
\(\dfrac{x^2 \times -\sin x - \cos x \times 2x}{(x^2)^2} = \dfrac{-x^2\sin x - 2x\cos x}{x^4} = \dfrac{-x\sin x - 2\cos x}{x^3}\) oe	A2,1,0
	(3)
	(5 marks)