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How to Find the Difference Quotient with Radicals Video

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  • 0:04 Difference Quotients…
  • 3:39 Solution
  • 4:03 Example
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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 review the difference quotient of functions and look at finding the difference quotient of functions with radicals. We will look at the steps involved in finding the difference quotient with radicals and before using those steps in examples.

Difference Quotients with Radicals

We are interested in finding the difference quotient of functions with radicals. You may be familiar with the formula for the difference quotient of a function f(x). That is:

[f(x + h) - f(x)] / h

where (x, f(x)) and (x + h, f(x + h)) are any two points on the graph of the function of f(x). This formula gives us the slope of a line through any two points on the graph of f. The area in which this difference quotient is most useful is in finding derivatives. You see, the limit of the difference quotient, as h approaches 0, is equal to the derivative of the function f.


diffquotrad1


Because of this, it's always desirable to be able to simplify the difference quotient of a function to a point where, if h = 0, there will not be a zero denominator.

Keeping this in mind, let's consider functions with radicals. When we want to find the difference quotient of a radical function, the first step is the same as if we were finding the difference quotient for any function. That is, we find f(x + h), and we plug f(x) and f(x + h) into the difference quotient formula.

For instance, consider the simplest radical function, f(x) = √(x). The first step in finding the difference quotient for this function is to find f(x + h) by plugging in x + h for x in f to get:

  • f(x + h) = √(x + h)

Now, we plug f(x) and f(x + h) into the difference quotient formula.


diffquotrad2


We end up with [ √(x + h) - √(x)] / h. Hmm. . . that looks like it's as far as we can take it, but if h = 0, there will be a zero denominator, so we really want to be able to take it further. We're going to have to get creative!

The next step in finding the difference quotient of radical functions involves conjugates. The conjugate of the expression a + b is a - b and the fact that

  • (a + b)(a - b) = a2 - b2

will really help us to simplify this difference quotient.

When we hit a roadblock that looks like we can't simplify the difference quotient that has radicals in it, the next step is to find the conjugate of the numerator and multiply both the numerator and denominator by that conjugate.

In our example, the numerator is √(x + h) - √(x), so the conjugate of this would be √(x + h) + √(x). Watch what happens when we multiply the numerator and denominator by this conjugate.


diffquotrad3


After all the simplification, we end up with 1 / [√(x + h) + √(x)] and we see that if h = 0, we don't get a zero denominator.

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