Write down \(f'(x)\).

Point \({\text{P}}(2,6)\) lies on the graph of \(f\).

Find the gradient of the tangent to the graph of \(y = f(x)\) at \({\text{P}}\).

Point \({\text{P}}(2,16)\) lies on the graph of \(f\).

Find the equation of the normal to the graph at \({\text{P}}\). Give your answer in the form \(ax + by + d = 0\), where \(a\), \(b\) and \(d\) are integers.

\(\left( {f'(x) = } \right)\)   \(4{x^3}\)     (A1)     (C1)

[1 mark]

\(4 \times {2^3}\)     (M1)

Note: Award (M1) for substituting 2 into their derivative.

\( = 32\)     (A1)(ft)     (C2)

Note: Follow through from their part (a).

[2 marks]

\(y - 16 =  - \frac{1}{{32}}(x - 2)\)   or   \(y =  - \frac{1}{{32}}x + \frac{{257}}{{16}}\)     (M1)(M1)

Note: Award (M1) for their gradient of the normal seen, (M1) for point substituted into equation of a straight line in only \(x\) and \(y\) (with any constant ‘\(c\)’ eliminated).

\(x + 32y - 514 = 0\) or any integer multiple     (A1)(ft)     (C3)

Note: Follow through from their part (b).

[3 marks]