If you have the equation for the tangent at the point where you want to find the instantaneous rate of change, then just look at the slope of the line. But if you only have two points on the tangent, then the method presented here will help you find the slope. You will see that we actually calculate the average rate of change to find the slope.
In this figure, you see the function (blue curve):
You draw the tangent (gray line) at the point (where means that the -coordinate of the point has the value , thus is a number). Note that one of the two points on the tangent line may be a point on the graph (the point of tangency).
You now choose a point you like on the tangent (gray line) and call it . Now, calculate the slope of the tangent to find the instantaneous rate of change.
You can use the expression for the average rate of change to find an approximation of the instantaneous rate of change at . Then you select an value very close to the -value you have been given in the exercise and enter both of them into the formula for the average rate of change.