How do you find the rate of change of y with respect to x?

Sep 23, 2014

The rate of change of y with respect to x, if one has the original function, can be found by taking the derivative of that function. This will measure the rate of change at a specific point. However, if one wishes to find the average rate of change over an interval, one must find the slope of the secant line, which connects the endpoints of the interval. This is computed by dividing the total change in y by the total change in x over that interval.

Given that this question was asked in the section on average rates of change, we shall discuss that possibility here. If you would prefer an answer to the other (the immediate rate of change at a point), place a question in that section, as this response is already going to be rather lengthy. Two examples for average rate of change shall be considered.

First, suppose that Otto is an exercise buff. Unfortunately, he is missing his pedometer. Otto decides to visit his friend, who lives four miles away. He decides to run the entire way there, and notes before he leaves that it is 5:15 PM. Upon his arrival, he notes that it is 5:55 PM, meaning that he has taken 40 minutes to make the run.

Using this, Otto can figure out his average velocity, to keep track of in his exercise log. We take his change in distance ($4$ miles), and divide by the change in the independent variable, time ($40$ minutes or $\frac{2}{3}$ hours). Dividing the change in distance by the change in time, one obtains an average velocity, or rate of change of distance with respect to time, of 0.1 miles per minute (or 6 miles per hour).

For our second example, consider a function $y = {x}^{2}$. Suppose one wants to know the average rate of change for this function over the inclusive $x$-interval $\left[2 , 5\right]$. To calculate this, we shall first calculate the value of the function at these points. ${5}^{2} = 25 , {2}^{2} = 4$, so $y \left(5\right) = 25 , y \left(2\right) = 4$

Now we calculate the change in $y$ divided by the change in $x$. $\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{25 - 4}{5 - 2} = \frac{21}{3} = 7$
The average rate of change in y with respect to x over the interval is 7; that is, for every single unit by which x changes, y on average changes by 7 units.