Notice that the slope of the tangent line will be horizontal at all of the maxima and minima of the graph.
Because the slope of a horizontal line is zero, this means that the derivative will be zero at those values
(± , ,...). Next, notice that the slope of the curve is about 1 as the curve goes through the origin. This
should make sense if you recall that = 1. The slope of the curve is about −1 as the curve goes
through x = ≠. And so on. If we now sketch the derivative, it looks something like the following:
Notice that this is the graph of y = cos x. This should be obvious because the derivative of sin x is cos x.
Now let’s do a hard one. Suppose we have the following graph: