Geographic Meaning of Differentiation

Hello All,

The following is a paragraph from my book...

"Find the slope of the tangent to the curve $y=\frac{1}{2x}+3$ at the point where $x=-1$ . Find the angle at which this tangent makes with the curve $y=2x^2+2$ . The slope of the tangent is the slope of the curve at the point where they touch one another; that is, it is the $\frac{dy}{dx}$ of the curve for that point. Here $\frac{dy}{dx}= -\frac{1}{2x^2}$ and for $x=-1, \frac{dy}{dx}=\frac{-1}{2}$ , which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation $y=ax+b$ , and its slope is $\frac{dy}{dx}=a$ , hence $a=\frac{-1}{2}$ . Also, if $x=-1, y=\frac{1}{(2)(-1)}+3=2\frac{1}{2}$ , and as the tangent passes by this point, the coordinates of that point must satisfy the equation of the tangent, namely: $y=-\frac{1}{2}x+b$ so that $2\frac{1}{2}=\frac{-1}{2}(-1)+b$ and $b=2$ ; the equation of the tangent is therefore $y=\frac{-1}{2}x+2$ "

So here are my questions regarding this paragraph:
1.) When I take the tangent of the first curve, I get $\frac{dy}{dx}=\frac{1}{2}$ ...how does he come up with $\frac{dy}{dx}=\frac{1}{2x^2}$ ?

(depending on the answer, I might have more questions to follow) Thanks.