Simplify the difference quotients ƒ(x+h) - ƒ(x) / h&nb...

Question

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = - (3/√x)

Accepted Answer

Welcome back, everyone. For the function F X equals -7 divided by root X, calculate the difference quotients F of X plus H minus F X divided by H and F X minus F A equals X minus A. Now, for this problem, basically, if we're going to find the different quotients, we have to substitute X + H and A respectively into our function F of X and then simplify. So let's go ahead and do that. I'm going to do F of X plus H minus F of X divided by H in blue and F of X minus F of A divided by X minus A in red, OK? So let's start with F of X plus H minus F X divided by H. We already know that F of X equals -7 divided by root X, so anywhere we see X, we're going to substitute X + H in F of X plus H. So this is going to be -7 divided by the square root of X + H, OK. Minus -7 divided by the square root of X, which is F of X, all divided by H. Now notice here we're subtracting a negative, so we could rewrite this then as plus. 7 divided by root X. Now let's simplify here by trying to combine the terms in our numerator. Now when we add both of these quotients. We're going to get it to be equal, OK, to -7. 7, root X plus 7 multiplied by root X plus H, OK. All divided by the square root of X multiplied by X + H. This is what happens when we multiply our denominators here. And all of this will be divided by H. If we think about dividing by H as multiplying by 1 divided by H, then we could write our H here, OK? So in other words, this is going to be negative 7 root X plus 7 root X plus H divided by H multiplied by root X multiplied by X plus H. Now, how can we make this any simpler? Well, notice our numerator has a command factor of 7. So if we factor 70, we're going to be left with root X plus H minus root X, OK. All divided by H root X, multiplied by X plus H, OK. And no, to simplify it even further, then we could multiply both sides of the fraction by the conjugate of the numerator, OK? In other words, we could multiply it by root X plus H plus root X in the numerator. And by the same in the denominator. Now we do that because then we can apply the difference of 2 squares in our numerator here, OK? So now, this is gonna be 7 multiplied by the square of root X plus H, which would just leave us with X + H minus the square of root X, which would just be X. And all of this is being divided by our denominator, which is gonna be H, root X multiplied by X plus H, OK. Multiplied by root X + H+ root X. Now when we have it like this, we know X minus X equals 0, and then we'll be left with 7 H and H divided by H equals 1. So now when we simplify this even further, then we're going to be left with 7 divided by square root of X multiplied by X + H multiplied by the square root of X + H plus root X. We can't simplify it any further, so this is going to be FFX plus H minus F of X all divided by H. Let's move on to the second part of our problem, the 2nd quotient. So now we want to find F of X minus F of A, all divided by X minus A. But again, what do we do? We already know that FF X is -7 divided by root X. FFA means anywhere we see A we substitute X. Anywhere we see X, we substitute A in our function. So that's gonna be -7 divided by root A. Now all of this is being divided by X minus A. And here, since we're subtracting a negative, we could rewrite this as + 7 divided by root A, OK. And no, if we factor out, if we combine both of these, OK, then this is gonna be equal to 7 multiplied by root X. OK, so multiplied by root X minus root A. So a factor 7 and combined both fractions, divided by root A X, OK. And then all of this is divided by X minus A, which we know then that we could rewrite this as 7 multiplied by root X minus root A, all divided by root AX multiplied by X minus A, similar to our previous problem. Now, where do we go from here? Well, we could factor root X minus root A from X minus A again using the difference of two squares because no. OK, we could, we have our numerator. But we could rewrite our denominator as root AX multiplied by root X plus root A multiplied by root X minus root A. And when we do that, no, we can cancel root X minus root A from the numerator and denominator. And no, that leaves us with 7 divided by root AX multiplied by root X plus root A. Therefore, this is F of X minus F of A divided by X minus A, and we had our previous answer for FX plus H minus F of X all divided by H. Thanks a lot for watching everyone. I hope this video helped.

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = - (3/√x)

Key Concepts

Difference Quotient

Rationalizing the Numerator

Function Evaluation

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