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 - 2ax +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 22 X plus 121, all divided by X minus 11. Where X is a variable and the denominator does not equal zero. Awesome. So it appears for this particular prompt we're asked to figure out what the derivative of F of X is using the quotient rule. So now that we know that we're ultimately trying to figure out what the derivative of F of X is using the quotient rule and noting that the denominator does not equal 0. Our first step in order to help us solve this problem is first off we need to determine or to figure out the derivative of the function. F of X is equal to X2 minus 22 X plus 121, all divided by X minus 11. And in order to do that, like the prone states, we will need to use the quotient rule. So we need to recall and note that the quotient rule states that F. Of X is equal to V multiplied by U. So when you take V. So once again, this is the quotient rule states that F of X is equal to V multiplied by U minus U multiplied by V all divided by V squad, and in this case, we need to note that for this particular problem, let's note. That you will be equal to X2 minus 22 X plus 121, and let us note that V in this particular problem will be equal to X minus 11. Awesome. So now, our next step is we need to differentiate you with respect to X, so you will be equal to X2 minus 22, so we need to take So now we're taking you, so we're differentiating you with respect to X, so we'll say that U is equal to X2 minus 22 X plus 121. So then we could say that U where U denotes the first derivative of U. So U is equal to 2 X minus 22. Fantastic. So now we need to differentiate V with respect to X. So V is equal to X minus 11, so V is equal to 1. So now our next step is we need to substitute in U V U and V into our quotient rule and state that D D D X D D X of X2 minus 22 X plus 121 divided by X minus 11. is all equal to when you factor out the when you factor out the numerator, you'll get that X minus 11 multiplied by 2 X minus 22. Minus And I said factor it out, but what I meant to say is you can factor it out later on to help make sure you cancel out all your like terms, but right now we're just substituting in our values for U V U and V into our expression. So once again, we got for for V we got X minus 11, and then for you we got X. Squared minus 22 X plus 121, etc. So, we need to take X minus 11 multiplied by 2X minus 22 minus X2 minus 22 X. Plus 121. And we need to multiply it by one. Awesome, so we need to take this multiply it by one. Fantastic. Then we need to divide everything by X minus 11 to the power of 2. So now we need to simplify our numerator. So when we do that, And let's write this in red. We'll get that X minus 11 multiplied by 2 X minus 22 minus X2 minus 22 X plus 121 is equal to 2. X to the power 2 minus 22 X plus 242242 minus X2. Plus 22 X. -121, and this is all equal to X2 minus 22 X plus 121. Awesome. Awesome. So now we need to sim, now we need to substitute in. Our known value back into our expression and so we need to substitute in our simplified numerator back into our equation. And when we do that, we will get that D by DX of X2 minus 22 X plus 121, all divided by X minus 11. It's going to be all equal to X2 minus 22 X plus 121. All divided by X minus 11 squad. Which when we factor out the top, you'll get that X minus 11 square divided by X minus 11 squad, which as you can see, those like terms cancel out, so that will mean that it's all equal to 1. Hooray, we did it. And that will mean that our final answer, without a doubt, and let's write it up above, that our fine actually we could squeeze it in down below. So we could say that our final answer without a doubt is F. of X is equal to 1. Where X cannot equal 11. So, without a doubt, our final answer once again is F of X is equal to 1, where X cannot equal 11. 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 - 2ax +a² / x-a, where a is a constant

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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