C3 June 2006 Q2

2. Differentiate, with respect to \(x\),

(a) \(\mathrm{e}^{3x} + \ln 2x\), (3)

(b) \((5 + x^2)^{\frac{3}{2}}\). (3)

Mark scheme (a)

Scheme	Marks
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x} + \dfrac{1}{x}\)	B1 M1 A1
	(3)

B1 \(3\mathrm{e}^{3x}\)

M1: \(\dfrac{a}{bx}\) A1: \(3\mathrm{e}^{3x} + \dfrac{1}{x}\)

Scheme	Marks
\((5 + x^2)^{\frac{1}{2}}\)	B1
\(\dfrac{3}{2}(5 + x^2)^{\frac{1}{2}} \cdot 2x = 3x(5 + x^2)^{\frac{1}{2}}\) M1 for \(kx(5 + x^2)^m\)	M1 A1
	(3)
	(6 marks)