Problem

A curve has equation

$$\begin{align*} y=2x^3-5x^2-\frac{3}{2x}+7, \qquad x>0. \end{align*}$$

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

The point $P$ lies on the curve and has $x$ coordinate $\dfrac12$.

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