Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-...

Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = (1/x) - x²

Accepted Answer

Hello there. Today we're going to 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. Simplify the difference quotient F of X minus F of A, all divided by X minus A for the function F of X is equal to 2 x 3 minus 1 divided by X. Awesome. So it appears for this particular problem we're ultimately trying to simplify the specific difference quotient for the function of F of X. So with that said, our first step that we need to do or need to take in order to solve this problem is we need to note that for our particular given difference function, we need to evaluate the required terms, so we need to note that F of X is going to be equal to 2 X 3 minus 1 divided by X, and we also need to note that F of A will be equal to 2, the power of 3 minus 1 divided by A. These we're plugging in a value of A in for X. Awesome. Our second step now is we need to substitute these values into our different quotient expression, so we need to write that F of X minus F of A, all divided by X minus A will be all equal to 2 X to the power of 3 minus 1 divided by X minus 2A of 3 minus 1 divided by A. All divided by X minus A. Fantastic. Our third step that we need to take is we need to note that at this stage the difference quotient needs to be simplified from this point on, so we need to first distribute the terms, so we need to state that F of X minus F of A, all divided by X minus A, will be equal to. 2 X to the power of 3, minus 2A of 3, minus 1 divided by X plus 1 divided by A. All divided by X minus A. Fantastic. So now our 4th step, which we at this point, we need to rewrite the numerator under a single fraction by using the common denominator A multiplied by X. So with that said, we need to take F of X minus F of A, all divided by X minus A, and it will be equal to 2 X to the power of 3 minus 2A of 3 minus 1 divided by X plus 1 divided by A. All divided by X minus A, and this will be all equal to 2 multiplied by A multiplied by X, all multiplied by X to the power of 3 minus A to the power of 3, all divided by a multiplied by X minus A divided by a multiplied by X, plus X divided by a multiplied by X. All divided by X minus A. And this will be all equal to 2 multiplied by A, multiplied by X, all multiplied by X to the power of 3, and when you take X to the power of 3 minus A to the power of 3, and we need to we need to subtract A, and we also need to add X. All divided by A multiplied by X, all multiplied by X minus A. Fantastic. So now, our 5th step We need to know that our goal is to ultimately obtain the factored form X to the power of 3 minus A power of 3 to get a common factor of X minus A. So we need to recall and use that A to the power of 3 minus B the power of 3 will be equal to A minus B multiplied by A2. Plus A multiplied by B plus B squared. So using this form, let us state that A is equal to X and B is equal to A for this particular case, following this formula. Awesome. So now, our next step will be at this point is we need to go ahead and write that. F of X minus F of A. All divided by X minus A will be equal to 2 multiplied by A, multiplied by X, multiplied by X minus A, multiplied by X squared. Plus A X plus a squared. Plus X minus A. All divided by A multiplied by X, all multiplied by X minus A. Fantastic. So now, our 6th step is we need to factor out the X minus A term, so we need to take F of X minus F of A, all divided by X minus A is equal to X minus A multiplied by 2. Multiplied by A, multiplied by X, all multiplied by X squared. Plus A X plus a squad, awesome. Plus one. And we need to divide everything by A multiplied by X multiplied by X minus A. And our 7th and final step is we need to cancel out the factor X minus A, and we also need to note that X minus A cannot equal 0. So when we do that, we will find that F of X, so we take F of X minus F of A, all divided by X minus A, is going to be equal to 2 multiplied by A multiplied by X, all multiplied by X2 + A X plus A2 + 1. All divided by A fantastic. So without a doubt, we could safely say that our final answer is As follows, So the first derivative of F of X, so the derivative of F of X, so we could say without a doubt, I should be specific here, that the difference quotient F of X minus F of A divided by X minus A for the function F of X is equal to 2 X of the power 3 minus 1 divided by X will be. 2 multiplied by a multiplied by X, all multiplied by X squared plus a multiplied by X, plus A to the power 2 + 1 all divided by a multiplied by X. And that is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = (1/x) - x²

Key Concepts

Difference Quotient

Function Composition

Limits

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