A derivative is the rate of change of a function, and is represented by the slope of the graph.

In question 1), the rate of change of distance with respect to time is SPEED.

To estimate the speed at t = 5, remember that the gradient/slope of a line is given by

.

At , .

At , .

So , , , .

So the slope (or estimate of the speed) is given by

.

For question 2), like I said, SPEED is the derivative of this function.

Supposing the distance between the two values of t was smaller. If this change in t (denoted by ) was closer and closer to 0, the approximation of the slope gets better and better.

So the limit is the limit as tends to 0 of the gradient formula.