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]