In the previous chapter, the evolutionary endpoint was one where the selection differential S ‘vanished’, which is jargon and means that it became zero: no further selection is then operating. The value of z that made S vanish was the one that made the derivative of fitness equal zero. Why did we calculate derivatives of fitness there? The derivative is something about how steeply a function increases (positive derivative) or decreases (negative derivative). This means that derivatives are very handy for finding out where bigger values of something can be found: just follow the uphill slope until … it turns negative, oops, now go no further. If we are dealing with a nice smooth function (there is some mathematical terminology for this, but just imagine something that bends gently without sharp edges), the ‘uphill’ part of the slope means a positive derivative (Fig. 4.1), and the point where uphill turns into downhill is where the derivative passes through zero on its way from positive to negative. In other words: where the derivative is zero, we have a good candidate for a local maximum of the function we were interested in.

So, the point where the selection differential S reached the value zero, also happened to maximize W. This is not a coincidence since S was indeed proportional to the partial derivative ∂W/∂z.