By seeing the name, "difference quotient formula", are you able to recollect something? The words "difference" and " quotient " are giving the feel of the slope formula.
Yes, the difference quotient formula gives the slope of a secant line that is drawn to a curve. What is a secant line? A secant line of a curve is a line that passes through any two points of the curve. Then the difference quotient of the function f x is shown below.
Thus using the slope formula , the slope of the secant line is,. The difference quotient formula is nothing but the slope of a secant line formula.
Since the difference quotient is nothing but the slope of a secant line, we use the slope formula to derive the difference quotient formula. This is the difference quotient formula. The difference quotient formula is mainly used to find the derivative. However, to make the process easier, it is best to take a good look at the given equation to make sure it is in the simplest form possible. Below is an example of what this means:.
As you can see, when trying to sub in terms into the difference quotient formula, the first equation will be much easier than the second. This is because the second equation involves a lot more operations, and the addition of "h" into that mix makes errors much more likely.
This will become more apparent as we get into some example problems. As it often goes, setting up the difference quotient is easiest for linear functions. For this kind of work, the less variables and the lower the degree, the easier it is! In order to use the difference quotient, all we need to do is set f a equal to f x , and make the necessary modifications to accommodate "h".
It is very important to keep track of what is in the parentheses, as this is where it is most common to make mistakes! Once we do all this, we simply substitute what we need from f x to set up the difference quotient. Once we have set up the difference quotient, all we need to do is expand whatever is in the brackets, collect like terms, and simplify! Polynomial functions are of similar difficulty when it comes to finding the difference quotient. Again, as the degree the highest exponent of the polynomial gets higher, as does the difficulty.
The most common difference quotient problems will deal with quadratic polynomial functions that have a degree of 2. Later, we will also look at a higher degree polynomial function by setting up a difference quotient for a cubic function. Again, in order to use the difference quotient, all we need to do is set f a equal to f x , and make the necessary modifications to accommodate "h".
Also remember it is very important to keep track of what is in the parentheses, as this is where it is most common to make mistakes! As an important side note, make sure you tackle these kinds of problem when the function is expanded out fully! If this function had been in factored form, it would have been much harder to set up the quotient. Just like last example, once we have set up the difference quotient, all we need to do is expand whatever is in the brackets, collect like terms, and simplify!
Just like with the quadratic function in the previous example, in order to use the difference quotient, all we need to do is set f a equal to f x , and make the necessary modifications to accommodate "h". This cannot be more important as we start dealing with more difficult equations and functions.
Finally, just like the last example, make sure you set up the difference quotient for the function in its fully expanded form, i. This is much more difficult compared to the quadratic example, as we have many more terms!
Be sure to double and triple check your work. It is very easy to lose track of terms in these kinds of questions. As you can imagine, dealing with even higher-order polynomials can be quite the challenge. We won't cover anything more difficult than cubic functions in this article, but it wouldn't hurt to challenge yourself! Practice makes perfect.
Polynomials and linear functions are often the simplest functions to deal with when doing problems on the difference quotient. The next sections cover rational and radical functions, which involve an extra level of difficulty. Even though we have a rational function, which is a bit more tricky, the technique is still the same. Since we are using a rational function, you need to pay extra attention to detail to make sure everything is set up correctly. Keeping in mind the careful attention we need to have, the procedure is still the same!
All we need to do is expand whatever is in the brackets, collect like terms, and simplify! This is the most basic rational function that exists, so, let's try something a little bit more difficult! Once again, even though we have a rational function, which is a bit more tricky, the technique is still the same. In this example, a slightly difference method is shown below. After the slightly different first step, the rest is the same. Keeping in mind the careful attention we need to have, all we need to do is expand whatever is in the brackets, collect like terms, and simplify!
Then, we can easily find our final answer. Last, but certainly not least, is finding the difference quotient for radical functions. By their very nature, radical functions are a bit more difficult than anything else. But, if we stay we the same technique we've been using since linear functions, nothing is all that different!
Just like with rational functions, don't let the tricky nature of radical functions fool you. The procedure is the exact same.
Also, since we are using a radical function, you need to pay extra attention to detail to make sure everything is set up correctly. Paying careful attention to the root signs and all of our terms, all we need to do is expand whatever is in the brackets, collect like terms, and simplify! And that's all there is to it!
In this article, we have gone over the basic steps to solve any difference quotient problem. But, your studying is not done there! Be sure to practice with even more complex functions to make sure you have a firm understanding of this topic. To aid in your practice using the difference quotient, check out this great calculator here. Good luck! Back to Course Index. You can still navigate around the site and check out our free content, but some functionality, such as sign up, will not work.
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