Why do you differentiate in optimisation questions?

When you differentiate a function it is giving you the rate of change of that function for different values of x. After subituting in, if it is positive it tells us that the graph is increasing, if it is negative it tells us it is decreasing and if it is zero it is neither increasing or decreasing: said to be stationary. If there is an optimal value of a function, small or large, this value can be found when the graph reaches a minimum or maximum turning point. These can also be called stationary points s the graph isn't going up or down; it is stationary. Setting the derivative of the function equal to zero gives us the values of x for any stationary points and substituting these back into the function we are then given the maximum or minimum value, which depending on the question could be an optimal value.