# 10.7 Derivatives and slopes

I really wasn’t sure what to call this section other than \sayderivatives and slopes.

Here’s something you’ve probably noticed before - if we have a straight line in the form $y=mx+c$, then if $m>0$ the line goes upwards (i.e. when $x$ increases, then $y$ increases too). If, however, $m<0$ then the line goes downwards (i.e. when $x$ increases, $y$ becomes smaller)!

Something very similar is the case for derivatives. Firstly, for a straight line99 9 Read: \sayThe derivative of $y$ with respect to $x$ is equal to $m$ $\frac{dy}{dx}=m$.

It is the case that for every function $f(x)$ (which we can differentiate) that whenever $\frac{dy}{dx}>0$ the function is increasing (i.e. as $x$ gets bigger, so does $y$). If $\frac{dy}{dx}<0$, then the function is decreasing (that is, when $x$ gets bigger, $y$ gets smaller).

## 10.7.1 Proof that the reciprocal function is decreasing

The reciprocal function1010 10 The function $f(x)=\frac{1}{x}$, can be differentiated,

 $\displaystyle\frac{d}{dx}\left[\frac{1}{x}\right]$ $\displaystyle=\frac{d}{dx}\big{[}x^{-1}\big{]}$ (10.45) $\displaystyle=-x^{-2}$ (10.46) $\displaystyle=-\frac{1}{x^{2}}$ (10.47)

Because $x^{2}$ is always greater than zero, it is also the case that

$\frac{1}{x^{2}}>0$ (10.48)

Therefore, $-\frac{1}{x^{2}}$ is always less than zero, so the reciprocal function is decreasing over its entire domain.