Derivatives Find and simplify the derivative of the following ...

Question

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the derivative of G of X E equals X + 2 multiplied by 3 X2 + 1, all divided by X cubed plus 8. For our answer choices, A says it's 2 multiplied by 3 X2 minus 11 X minus 1, all divided by X2 minus 2 X + 4 squared. B-2 divided by 3 X2 minus 11 x minus 1 divided by X2 minus 2 X + 42. C 2 multiplied by 3 x + 11 X minus 1 divided by X2 + 2 x + 4 squad, and the D negative 2 divided by 3 x 2 + 11 X minus 1 divided by X2 + 2 X + 4 quad. Now, how can we determine our derivative here, G of X? Well, we could start by using the caution rule of differentiation, but before we do that, we have an opportunity to simplify our expression. If we do that, it'll make it a little bit easier to solve. Now notice in our denominator here X cubed plus 8, it can be factored as a sum of cubes. Recall, the sum of cubes basically tells us that if we are adding two cubes, OK, we could rewrite it as. A plus B, where our cubes are a cubes plus B cube, multiplied by A2 minus AB + B2. OK? Now, in this case, we know that. If we apply this rule, then if we take a X cubed plus 8. It's really X cubed plus 2 cubed, OK, because it is 2 cubed. And if we use our sum of cubes, then we could write this as X + 2 multiplied by X2 minus 2 X + 4, OK? Using A to be X and B to be 2 from our sum of cubes formula. So what does this mean? Well, no, that means we could rewrite the effects. As X + 2. Multiplied by 3 x2 + 1, all divided by X + 2 multiplied by X2 minus 2 X + 4. Now we can factor out X + 2 and G of X is gonna be 3 X2 + 1 divided by X2 minus 2 X + 4. So we've we've written GFX a little bit more simply. No, we can still apply the quotient rule. What does the quotient rule say? Well, the quotient rule also tells us that by the quotient rule. It tells us that if we have some function Y that's a quotient, you divided by V where U and V are functions of X, then the derivative of Y is going to be equal to the sorry, is going to be equal to V multiplied by the derivative of U minus U multiplied by the derivative of V all divided by V2, OK. So if we identify you as our numerator here in this problem 3X2 + 1 and V as our denominator, X2 minus 2X + 4, then we can differentiate them and simplify to get the derivative of G of X. So let's go ahead and try to do that, OK? So first, let's let you be equal to 3 X2 + 1, OK? And V be equal to X2 minus 2 x + 4. If U is 3X2 + 1, then that means the derivative of U is going to be equal to 6 X, OK, by the power rule. And the derivative of V is gonna be equal to by the power rule 2 X. The derivative of X is 2 X minus 2, the derivative of negative 2 X is -2. Plus 0, the derivative of 4 is 0, which would just leave us with 2 x minus 2. Now that we have UV, and its derivatives, we can apply the quotient rule. So now, by applying the quotient rule, then we should get the derivative of G of X to be equal to V. X2 minus 2 X + 4 multiplied by U 6 X. Minus You, which is 3x2 + 1 multiplied by. Zebra, which is 2 X minus 2. I know all of this, OK, all of this is being divided by X2 minus 2 X + 4 all squared. No, if we go ahead and try to simplify our numerator, OK, then we can expand our expression here. So that's gonna be 6 X cubed minus 12 X squared. Plus 24 X. Minus, if we expand our, the product of our two terms here, it's gonna be 6 X cubed minus 6 X squared plus 2X minus 2, and divided by the same denominator, OK? Now if we think about it here, if we remove our bracket, we're subtracting this entire expression. So that's gonna be minus 6 X cubed plus 6 X squared. OK, minus 2 X. I'll just put it in black to show that I, I've changed it, plus 2, OK, minus -2 becomes +2. And now that we've removed our bracket, it makes it a little easier to read to simplify, OK? Because no, 6 XQ minus 6 X cubed is 0, OK. Uh, we're gonna have negative 12 X2 + 6X2. So that's gonna be -6X2. 24 X minus 2 X, that's plus 22 X. And then we're left to X + 2, OK? All of that is divided by our denominator. And then we could take it even a step further by factoring out 2, OK? Or factoring out -2. When we do that, we'll have a negative 2 multiplied by 3 X squared, minus 11 X minus 1. That's all divided by X2 minus 2 X + 4, all squared, OK? So this is gonna be the derivative of G of X. If we look back at our answer twice, that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)

Key Concepts

Derivatives

Product Rule

Quotient Rule

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