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

*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 (

For our second example, consider a function

Now we calculate the change in

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.

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To find the rate of change of ( y ) with respect to ( x ), you can use calculus. Specifically, you would take the derivative of the function ( y ) with respect to ( x ). This derivative, often denoted as ( \frac{{dy}}{{dx}} ) or ( y' ), represents the rate of change of ( y ) with respect to ( x ) at any given point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the average value of the function # f(x)=(x-1)^2# on the interval #[1,5]#?
- What is the average rate of change of the function #f(x)=2x^2 -3x -1# on the interval [2, 2.1]?
- How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrt(4x-5)#?
- What is the equation of the tangent line of #f(x)=sqrt(x+1)-sqrt(x+2) # at #x=2#?
- How do you find the derivative of # e^(xy)=x/y#?

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