Question

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

Answer 1

Hint: To find the rate of change of $ y $ with respect to $ x $ , we should understand first, what means by the rate of change of any variable with respect to any variable. We will consider a function of one variable with respect to another and then derivate it in it’s manner.

Complete step-by-step answer:
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 pace of progress at a particular point. In any case, on the off chance that one wishes to locate the normal pace of progress over a span, one should discover the slant of the secant line, which interfaces the endpoints of the stretch. This is processed by isolating the absolute change in y by the all out 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. On the off chance that you would lean toward a response to the next (the quick pace of progress at a point), place an inquiry in that segment, as this reaction is as of now going to be somewhat long.
Now,
Consider a function $ y = {x^2} $ . Suppose one wants to know the average rate of change for this function over the inclusive $ x - \operatorname{int}erval[2,5] $ . To calculate this, we shall first calculate the value of the function at these points.
 $ {5^2} = 25,{2^2} = 4;\,so\,y(5) = 25,\,y(2) = 4 $
Now we calculate the change in $ y $ divided by the change in $ x $ .
 $ \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{25 - 4}}{{5 - 2}} = \dfrac{{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.

Note: At its simplest, the rate of change of a function over an interval is just the quotient of the change in the output of a function $ (y) $ over the difference in the input of the function $ (x) $ \[\left( {change{\text{ }}in{\text{ y}}/change{\text{ }}in{\text{ x}}} \right)\] . More specifically, for any function \[f{\text{ }}\left( x \right)\] , the average rate of change of that function over the interval \[a{\text{ }} \leqslant {\text{ }}x{\text{ }} \leqslant {\text{ }}b\;\]