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Question: How do we find the rate of change of \( y \) with respect to \( x \) ?...

How do we find the rate of change of yy with respect to xx ?

Explanation

Solution

Hint : To find the rate of change of yy with respect to xx , 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=x2y = {x^2} . Suppose one wants to know the average rate of change for this function over the inclusive x−int⁡erval[2,5]x - \operatorname{int}erval[2,5] . To calculate this, we shall first calculate the value of the function at these points.
52=25,22=4; so y(5)=25, y(2)=4{5^2} = 25,{2^2} = 4;\,so\,y(5) = 25,\,y(2) = 4
Now we calculate the change in yy divided by the change in xx .
y2−y1x2−x1=25−45−2=213=7\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{25 - 4}}{{5 - 2}} = \dfrac{{21}}{3} = 7
The average rate of change in yy with respect to xx 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)(y) over the difference in the input of the function (x)(x) (change in y/change in x)\left( {change{\text{ }}in{\text{ y}}/change{\text{ }}in{\text{ x}}} \right) . More specifically, for any function f (x)f{\text{ }}\left( x \right) , the average rate of change of that function over the interval a ⩽ x ⩽ b  a{\text{ }} \leqslant {\text{ }}x{\text{ }} \leqslant {\text{ }}b\;