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Calculus: Help and Review13 chapters | 148 lessons

Instructor:
*Kimberlee Davison*

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn about the relationship between the first derivative and rates of change. You will learn how to find the derivative of a polynomial using limits.

The **first derivative** of a function is a new function (equation) that gives you the instantaneous rate of change of some desired function at any point.

Suppose you are playing a video game. You move your character, Squirmy, through a long underground tunnel using your control pad. One thing the pad allows you to do is to speed up and slow down. So, Squirmy sometimes travels faster and sometimes travels slower through the tunnel.

Maybe you are impatient to get Squirmy through the tunnel, so you use the controls to gradually speed him up. His speed at any point in time looks like this:

*s*= 2*x* + 5

In this equation, *x* represents the number of seconds since Squirmy entered the tunnel and *s* represents his speed (in cm/second). If you put a 0 in for *x*, you see that at time zero (when he entered the tunnel), he was traveling at 5 cm/second. One second later (*x* = 1), he was traveling at 7 cm/sec. After another second (*x* = 2), he was moving at the speed of 9 cm/second. Clearly, Squirmy is picking up speed. His instantaneous rate of change (speed at one instant in time) is constantly changing.

An equation that gives us the rate of change at any instant is a first derivative. If *y* is the distance, or location, then we usually label it *dy/dx* (change in *y* with respect to *x*) or *f ' (x)*.

We already have Squirmy's first derivative function, but let's pretend for a moment we don't. Pretend that all we have is a function that tells us where he will be at any instant. In this case, we might have:

*y* = *x*^2 + 5*x*, where *y* is Squirmy's distance from the entrance of the tunnel.

The formula for the first derivative is really just based on the idea of relative change. All we want to know is at some instant how quickly his distance (*y*) is changing with respect to time (*x*).

- Speed at a specific time = instantaneous rate of change at a specific time = first derivative at a specific time = (change in
*y*)/(change in*x*)

One complication here is that the 'change in *x*' (time) at an instant is zero. So, in theory, we need to divide by zero. Luckily, we have some ways around that.

The way around that is we will not evaluate (change in *y*)/(change in *x*) when the change in *x* is zero. Instead, we will look what happens to that ratio (fraction) as change in *x* gets very close to zero. In other words, we use the concept of **limits** from calculus.

Mathematically, you write it this way:

So, you find how far he travels in an instant, divided by the length of that instant, and then you find what that gets really close to as the instant gets smaller and smaller (closer to zero).

Using Squirmy's distance function, *y* = *x*^2 + 5*x* or *f(x)* = *x*^2 + 5*x*, you first find where he will be an instant (*h*) from now:

*f*(*x* + *h*) = (*x* + *h*)^2 + 5 (*x* + *h*) = *x*^2 + 2*xh* + *h*^2 + 5*x* + 5*h*

Next, you just plug values into the formula and use some algebra.

This should look a lot like Squirmy's speed function from the beginning of the article.

Finding the first derivative of a polynomial always looks a lot like this. *h* will always divide into the numerator, getting rid of a divide-by-zero problem.

Suppose now that you have a different polynomial function:

*f*(*x*) = 2*x*^3 + 2

Finding the derivative involves the same steps.

First, find *f*(*x* + *h*) = 2(*x* + *h*)^3 - 3(*x* + *h*) + 2 = 2(*x*^3 + 3*x*^2*h* + 3*xh*^2 + *h*^3) +2

Next, plug into the formula for the derivative and simplify.

The first derivative is a formula for the instantaneous rate of change of one variable with respect to another. Using the limit formula, finding the first derivative of a polynomial requires simply following a set of predictable steps.

Gather information about the first derivative via this lesson, then do your best to:

- State the definition of a first derivative
- Write formula for solving for the first derivative
- Find a first derivative by following the correct steps

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Calculus: Help and Review13 chapters | 148 lessons

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