# Derivatives

### Topics

A **rate** is a value that expresses how one quantity changes with respect to another quantity. For example, a rate in "miles per hour" expresses the increase in distance with respect to the number of hours we've been driving.

If we drive at a constant rate, the distance we travel is equal to the rate at which we travel multiplied by time:

Dividing both sides by time, we have

### Sample Problem

If you drive at 50 mph for two hours, the distance we'll travel is 50 mph × 2 hrs = 100 miles.

### Sample Problem

If we drive at a constant speed for 3 hours and travel 180 miles, we must have been driving

.

In real life, though, we don't drive at a constant rate. When we start our trip through Shmoopville, we first climb into the car, traveling at a whopping 0 miles per hour. We speed up gradually (hopefully), maybe need to slow down and speed up again for traffic lights, and finally slow down back to a speed of 0 when reaching our destination, The Candy Stand. We can still divide the distance we travel by the time it takes for the trip, but now we'll find our **average rate**:

To calculate the average rate of change of a dependent variable *y* with respect to the independent variable *x* on a particular interval, we need to know

- the size of the interval for the independent variable, and

- the change in the dependent variable from the beginning to the end of the interval.

Depending on the problem, we may also need to know

- the units of the independent and dependent variables.

Then we can find

The average rate of change of *y* with respect to *x* is the slope of the secant line between the starting and ending points of the interval:

Relating this to the more math-y approach, think of the dependent variable as a function *f* of the independent variable *x*. Let *h* be the size of the interval for *x*:

and let *a* be one endpoint of the interval, so the endpoints are *a* and *a* + *h*, with corresponding *y*-values *f*(*a*) and *f*(*a* +* h*):

Then the slope of the secant line is

This is the definition of the slope of the secant line from (*a*,* f*(*a*)) to (*a* + *h*, *f*(*a* + *h*)).