C4 June 2010 Q3

3. A curve \(C\) has equation \[2^x + y^2 = 2xy\]

Find the exact value of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) at the point on \(C\) with coordinates \((3, 2)\). (7)

Mark scheme

Scheme	Marks
\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(2^x\right) = \ln 2 \cdot 2^x\)	B1
\(\ln 2 \cdot 2^x + 2y\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2y + 2x\dfrac{\mathrm{d}y}{\mathrm{d}x}\)	M1 A1= A1
Substituting \((3, 2)\) \(8\ln 2 + 4\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4 + 6\dfrac{\mathrm{d}y}{\mathrm{d}x}\)	M1
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4\ln 2 - 2\) Accept exact equivalents	M1 A1
	(7)
	(7 marks)