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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will show how to find the derivative of the square root of x. We will then look at two different ways to check our work: one involving limits and the other involving integrals.

We want to find the derivative of the square root of *x*. To get started, we need to be aware that the square root of *x* is the same as *x* raised to the power of 1/2. In general, we know that the *n*th root of *x* is equal to *x* raised to the power of 1/*n*. Since the square root of *x* is the second root of *x*, it is equal to *x* raised to the power of 1/2.

You may be wondering why we want to think of the square root of *x* in this way. Well, as it turns out, we have a nice formula we can use to find the derivative of *x**a*.

Thus, if we think of the square root of *x* as *x* 1/2, then we can use the formula to find the derivative.

The formula gives that the derivative of the square root of *x* is (1/2)*x* -1/2. This can be written in a few different forms:

There are a couple different ways that we can check our work when dealing with derivatives. The first deals with the definition of a derivative using limits.

We can use this definition to check our work. In doing so, we should get the same result as we did when using the formula. We start by letting *f*(*x*) = sqrt(*x*), and we plug in accordingly.

Now we want to find the limit as *h* approaches 0. One way of evaluating a limit is by plugging the number that *h* is approaching in for *h*. However, in this case, we would be plugging in 0 for *h*. Can you see why we can't do that? If you are thinking that we can't plug in 0 for *h* because that would create a zero denominator, then you are correct! Therefore, we are going to manipulate the limit to get it into a form where we can plug in 0 for *h* without creating an undefined expression. We will multiply it all by a version of the number 1:

Remember, we didn't change the limit since we ultimately just multiplied it by one. Also, notice that we now are able to plug in zero for *h* without creating a zero denominator or an undefined expression. Let's do just that to find the limit and, in the process, find the derivative of the square root of *x*. Once we plug 0 in for *h*, our equation becomes:

See, the derivative of the square root of *x* is (1/2)*x* -1/2, which is exactly what we got when we used the formula. Phew! This is good news! It means that we did our work correctly.

Another way to check our work is by using integrals. **Integrals** are called anti-derivatives, and they basically undo derivatives. That is, if *a* is the derivative of *b*, then the integral of *a* is *b* + *C*, where *C* is a constant.

This tells us that in our example, since the derivative of sqrt(*x*) is (1/2)*x* -1/2, it should be the case that the integral of (1/2)*x* -1/2 is sqrt(*x*) + *C*, where *C* is a constant. You may not be familiar with integrals yet, but that's okay. We are lucky enough to have two easy to follow facts that will allow us to find the integral of (1/2)*x* -1/2.

1.) The integral of a constant times a function is that constant times the integral of the function.

2.) The formula for the integral of *x* *n* is equal to:

Using these two rules, we can find the integral of (1/2)*x* -1/2, and verify that it is sqrt(*x*) + *C*, where *C* is a constant. This will allow us to check that we did our work correctly.

Just as we hoped, we see that the integral of (1/2)*x* -1/2 is sqrt(*x*) + *C*, where *C* is a constant. Great! Once again, our work checks out.

When working with derivatives, both the function of derivatives using limits and integrals are extremely useful for making sure that we did our work correctly.

Study the lesson thoroughly and retain enough information to confidently:

- Solve for the derivative of the square root of x
- Utilize integrals to check one's work

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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