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In this lesson, you will learn about the instantaneous rate of change of a function, or derivative, and how to find one using the concept of limits from Calculus.

What Is Instantaneous Rate of Change?

An instantaneous rate of change, also called the derivative, is a function that tells you how fast a relationship between two variables (often x and y) is changing at any point.For example, suppose you have taken up archery lessons. When you shoot your arrow, it leaves your bow quickly and then gradually slows down until it hits the target on the other side of the field. At first, you aren’t very good.

By the time the arrow hits the target, it has slowed down so much that it bounces off the target without even piercing it. If your pet pony, Bubbles, happened to trot across the lawn at just the wrong moment, she wouldn’t really be in any danger. Bubbles would barely feel the arrow bounce off her.Over time, you get better. The arrows leave your bow with much greater speed. While your aim might not be much better, the arrows do pierce the target now because they are still traveling quickly when they hit it. Now you have to be a little more careful to keep Bubbles out of the yard when you practice.

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The difference in your shooting is the instantaneous rate of change when the arrow hits the target (or Bubbles). It is the speed at which the arrow is traveling at the instant when it makes contact. Obviously, if the arrow is moving at 0 feet per second, it isn’t going to hurt Bubbles, your neighbor’s dog, or anything else. The instantaneous rate of change is zero.

The faster it is moving at the time of contact, however, the worse danger Bubbles is in.If your arrow gradually slows down after it leaves your bow, then the distance to the target matters. If Bubbles crosses the yard at a distance of 2 yards from you, then she is in worse danger than if she crosses 100 yards away – assuming you accidentally get a direct hit either way. The speed when the arrow hits – the instantaneous rate of change – is what matters.

The Derivative and Slope

In mathematics, we often think of instantaneous rate of change as a slope.

You may remember that the slope of a line can be thought of as rise/run, or the change in y divided by the change in x.Slope = (change in y)/(change in x)One problem with instantaneous rate of change is that it refers to just one instant in time – it is the change in y happening when the change in x is just an ‘instant,’ technically zero.If you put zero into the denominator of the slope formula, however, you have a problem.

Dividing by zero will cause the universe to implode (or something else equally horrific that math teachers keep as a secret from the rest of the world). So, what do you do?Well, the easiest method is to use limits from calculus. Instead of putting a zero in the denominator directly, you ask what happens to the slope as the zero gets closer and closer to zero.In the archery example, that means you look at how much distance the arrow travels over a very small amount of time – and then you make that very small amount of time smaller and smaller until it is almost zero. If we call that small amount of time, h, then the formula for instantaneous velocity looks like this:

Parabola with a tangent line at (1,3)

Now, the slope of the blue tangent line is the instantaneous velocity at that point. So, how do you find it? That’s where limits come in.

Pretend that instead of one point (black dot) on the curve, there are two black dots very close to each other. In fact, the x-values of those two black dots are h apart. The y-values are this much apart: f(x+h) – f(x).

In that expression f(x) is the y-value at the first black dot. f(x+h) is the y-value at the next black dot – at h more than x.So, the slope of the line connecting the two black dots is:f(x+h) – f(x)/h

If h happens to be zero, then the two points are right on top of each other (as is really the case in the picture).Again, you can’t divide be zero, so instead you use limits. You get:

Example worked through

The instantaneous rate of change, or derivative, can be written as dy/dx, and it is a function that tells you the instantaneous rate of change at any point. For example, if x = 1, then the instantaneous rate of change is 6.

Lesson Summary

The instantaneous rate of change tells you how fast y is changing with respect to x at any value of x. It is also called the derivative, and it is the slope of the line tangent to a graph at any point.

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