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 , then if the line goes upwards (i.e. when increases, then increases too). If, however, then the line goes downwards (i.e. when increases, becomes smaller)!
Something very similar is the case for derivatives. Firstly, for a straight line99 9 Read: \sayThe derivative of with respect to is equal to .
It is the case that for every function (which we can differentiate) that whenever the function is increasing (i.e. as gets bigger, so does ). If , then the function is decreasing (that is, when gets bigger, gets smaller).
10.7.1 Proof that the reciprocal function is decreasing
The reciprocal function1010 10 The function , can be differentiated,
Because is always greater than zero, it is also the case that
Therefore, is always less than zero, so the reciprocal function is decreasing over its entire domain.