# 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
line^{9}^{9}
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 function^{10}^{10}
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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

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