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) = x⁴

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 3 x to the power of 3. Awesome. So it appears for this particular prompt, we're trying to simplify this difference quotient for the function where F of X is equal to 3x the power of 3. So now that we know what we're solving for, our first step that we need to take is we need to note that for our particular given like difference quotient, we need to evaluate our required terms, so we need to focus on F of X. is equal to 3 X to the power of 3, and we also need to note and recall that F of A. It's going to be equal to 3A to the power of 3, means we're using a value of A X. Our second step now is we need to substitute these values into our different quotient expression, so we need to take F of X minus F of A, all divided by X minus A, and this will be equal to 3 x 3 minus 3A to the power of 3, all divided by X minus A. Now our third step now is we need to focus our attention now on the difference quotient, and the difference quotient needs to be simplified from this point. So we need to first, in order to simplify, we need to first factor out the leading coefficient. So we need to take F of X minus F of A. Divided by x minus A is equal to 3 x 3 minus 3A 3, all divided by X minus A, which will be equal to 3 multiplied by X 3 minus A 3, all divided by X minus A. Fantastic. So now our 4th step is we need to Use the differentiation of cubes. factorization. So we're now trying to use the difference of cubes factorization, as we should recall how to do that, such that, and let's make a note of this, such that A to the power of 3 minus B the power of 3 is equal to A minus B multiplied by A2 plus A multiplied by Bplus B squad. Fantastic. So with that in mind, let us say that A equals X, and let us say that B equals A in this particular case. So when we do that, we could go ahead and write that X 3 minus A 3 will be equal to x minus A multiplied by X2 + A X plus a squad. Awesome. So now, For our 5th step, we need to substitute what we know, what we've what results we found into the expression that we determined in step 4, into the form that's given in step 3. So when we do that, we will take F of X. So when you take F of X minus F of A. All divided by X minus A will be equal to 3 multiplied by X minus A multiplied by X2 + A X plus A2. Fantastic, and then we need to divide everything by X minus A. Fantastic. So now what we need to do. We need to cancel out the common factor, and let us also make a note here off to the side, so we're canceling out the common factor, and we also need to note that X minus A cannot equal 0, so we could go ahead and write that for step 6, that F of X minus F of A, all divided by X minus A. We could say that this is equal to. When we can once again, when we cancel out the common factor, it's equal to 3 multiplied by X2 plus. AX Plus 8 to the power of 2, and that is our final answer. Hooray, we did it. So without a doubt. In conclusion, 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 3x the power of 3. In its factored form will be F of X minus F of A, all divided by X minus A will be equal to 3 multiplied by X2 plus A X plus A to the power of 2 or A2. 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) = x⁴

Key Concepts

Difference Quotient

Polynomial Functions

Limit and Derivative

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