7–14. Find the derivative the following ways:
a. Using t...

Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

y = x² - a² / x-a, where a is a constant

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of the function using the quotient rule. F of X is equal to X2 minus 121, all divided by X minus 11, where X is a variable and the denominator does not equal 0. Awesome. So it appears for this particular problem, we're asked to figure out what the derivative of the function of FFX is using the quotient rule. So now that we know that we're trying to figure out the derivative of FFX using the quotient rule, First off, what we need to do in order to help us solve this particular problem. we need to identify the numerator and denominator. The numerator in this case will be U is equal to X2 minus 121, and the denominator, which we'll call V and V is equal to X minus 11. So now, our next step is we need to differentiate the numerator and denominator. The derivative of the numerator will be U and U is equal to 2 X, and the derivative of the denominator, which will be V is equal to 1. And now we need to recall and apply the quotient rule, and as we should recall, the quotient rule states that F of X is equal to V multiplied by minus U multiplied by V. All divided by V2. So now what we need to do is we need to substitute in our values of U, U, V, and V into the quotient rule, and when we do that, we will find that F of 1, so we need to say that F I said F of 1, but I meant F of X. So F of X. is equal to x minus 11. And we need to multiply this by 2X minus X squad minus 121 multiplied by 1. All divided by X minus 11 to the power of 2, and once again this is when this is all. All we're doing is we're plugging in our known values for U, V, and U and into our equation for the quotient rule. So now we need to simplify our expression. So when we Expand and simplify the numerator, we will find that F of X is equal to 2 X2 minus 22 X minus X2 + 121, all divided by X minus 112. And now when we combine like terms, we could write that F of X is equal to X2 minus 22 X plus 121, all divided by X minus 112, fantastic. And we need to factor the numerator at this stage if it is possible. So we need to notice at this point, as we should recall a note, that the numerator in this case is a perfect square trinomial. So that means we could write that F of X is equal to X minus 11 2 divided by X minus 11 2. So we could see in this case that since the numerator and denominator are the same, we could go ahead and write that F of X is equal to 1. Where X cannot equal 11. So therefore, without a doubt, the derivative of the function F of X is equal to X2 minus 121, all divided by X minus 11 is F of X is equal to 1, where X cannot equal 11, and that is our final answer. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - a² / x-a, where a is a constant

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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