Rgba,

Slope is rate of change. The "average" rate of change is the slope of the function, while the "instantaneous" rate of change is the derivative; the slope of the line tangent to some point of the function.

Consider the following, hypothetical scenario and problem:

"A piece of metal falls from a rocket that's three hundred feet in the air. The height of the metal is proportional to three times the square of the time, in seconds, of which the piece of metal has fallen. What is the average speed of the piece of metal at impact with the ground and what is its instantaneous speed?"

To find its "average" speed we must write a function to express the height of the ball. The function will be

, with respect to time

, that is

. To find its average rate of change (speed) we must find the time at

, which can be solved through the following quadratic equation:

Now, to find the average speed at impact we will divde the output of

by the input,

. Therefore, the average speed of the piece of metal is 30 feet per second (the negative is due to the fact that the distance traveled is negative).

In order to find the instantaneous rate of change (the instantaneous speed) we must differentiate the function, which is a polynomial, which means we must differentiate every term of the polynomial.

The derivative of the function

is

. Now, plugging in h'(10) we get the instantaneous velocity, which is

, or 60 feet per second.

I hope this has helped you to some extant to understand what the derivative is and one of its many applications.