Problem

The curve $C$ has equation

$$\begin{align*} y=\frac{x^2}{3}+\frac4{\sqrt{x}}+\frac8{3x}-5, \qquad x>0. \end{align*}$$

(a) Find $\dfrac{dy}{dx}$, giving your answer in simplest form.

The point $P(4,3)$ lies on $C$.

(b) Find the equation of the normal to $C$ at the point $P$. Write your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.