Let \(f(x) = \frac{{g(x)}}{{h(x)}}\), where \(g(2) = 18,{\text{ }}h(2) = 6,{\text{ }}g'(2) = 5\), and \(h'(2) = 2\). Find the equation of the normal to the graph of \(f\) at \(x = 2\).

recognizing need to find \(f(2)\) or \(f'(2)\)     (R1)

\(f(2) = \frac{{18}}{6}\)   (seen anywhere)     (A1)

correct substitution into the quotient rule     (A1)

eg     \(\frac{{6(5) - 18(2)}}{{{6^2}}}\)

\(f'(2) =  - \frac{6}{{36}}\)     A1

gradient of normal is 6     (A1)

attempt to use the point and gradient to find equation of straight line     (M1)

eg     \(y - f(2) =  - \frac{1}{{f'(2)}}(x - 2)\)

correct equation in any form     A1     N4

eg     \(y - 3 = 6(x - 2),{\text{ }}y = 6x - 9\)

[7 marks]