Finding Derivative Values

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Question

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

Accepted Answer

Welcome back, everyone. Let F of X equal 2 X minus 3 divided by 2 X + 1 all cubed, and the G of X equals 4 divided by X2 minus 1. Find the value of F of G of -2. A says it's 24, B-24, C 216, and D-216. Now what are we trying to figure out here? Well, essentially, F of G of -2 is the derivative of the composite function F G X evaluated at X equals -2. So since we're talking about composite functions, we'll need to use the chain rule of differentiation. Recall, it basically tells us that the derivative of a composite function F of G of X. is equal to F of G of X, OK. The derivative of F of G of X multiplied by G of X, the derivative of G of X. So if we apply that idea to our problem. Then this means that F of G of -2. Is going to be equal to F of G of -2, OK, multiplied by the derivative of G-2. OK. So there are a few things that we'll need to find that we want to plug into our function or plug into our derivative to help us solve for F of G of negative2. First, let's start by finding G of negative2. Now, G of -2, OK, we have G of X 4 divided by X2 minus 1. So this is gonna be 4 divided by -2 2 minus 1. -2 squad is 44 divided by 4 is 1, and 1 minus 1 equals 0, OK? So G of -2 is 0. Next we need to find the derivative of G of X. Now, here, for the derivative of G of X, OK. That means we're gonna be finding the derivative of 4 divided by X2, which you can write as 4 X to the negative2 minus 1. Now in this case. Then this is gonna be equal to 4 multiplied by -2, multiplied by x -2 minus 1, the derivative of 4 X 2 minus 0, the derivative of 1, because 1 is a constant. Now when we simplify it here, this is gonna be -8 X to the negative cubed, which is gonna be the negative value of a divided by X cubed. This is the derivative of G of X and in that case then this means that G of -2 is then gonna be equal to -8 divided by -2 cubed. That's -8 divided by -8, which equals 1. So we found the G of -2. Now we need to find F G negative2. Before we can do that, we need to find F of X. We need to differentiate F of X. Now remember, we know that F of X is equal to 2 X minus 3 divided by 2X + 1 are cubed. So it's also a composite function. So we're gonna bring down our power 3 multiplied by 2 X minus 3 divided by 2 X + 1. OK, raise to the power of 3 minus 1. OK, we've differentiated the other function, and now we're gonna multiply by the derivative of the inner function. That is the derivative of 2 X minus 3 divided by 2 X + 1. Now, this function is a quotient. So to differentiate it, we can use the quotient rule, OK? Now, for the quotient rule, basically recall that and let's just make a box over here to show that differentiation, OK? So now, OK, to differentiate it. If we let you be equal to 2 X minus 3 and V be equal to 2 X + 1. And differentiate both of these, the derivative of you would be 2 and the derivative of V would also be 2. Then by the quotient rule we're going to. Well, Let me write, I thought. If Y equals U divided by V, then its derivative would be equal to Vu minus UV divided by Vquared. So we're gonna plug this into our expression, OK? We're gonna plug this into our quotient here to find the derivative of 2 X minus 3 divided by 2 X + 1. So now if we think about it here. If we do a differentiation, OK, if we, if we plug in the quotient rule, then this means the derivative is gonna be equal to 2 X + 1 multiplied by 2. OK, minus 2X minus 3 multiplied by 2, all divided by the original denominator 2 X + 1 all squared. I know if we do some uh simplification here. 2X plus 1 multiplied by 2, that's 4 X plus 2, minus 2 X multiplied by +24 X minus 3 multiplied by 2 minus 6. OK. Again, all of this is divided by 2 x + 1 are squared. And no, we could continue to simplify, but I'm just gonna bring it over to our problem here. So no, if we put it into our problem, that's 3 multiplied by 2X minus 3 multiplied by 2X + 1 squared, OK, multiplied by our derivative here by the quotient rule, which is gonna be 4 X + 2 minus 4 X + 6, OK? All divided by 2 X + 1 squared. I know if we go ahead and simplify here, OK, then this is gonna be 3 multiplied by 2X minus 3 squared, OK, multiplied by, well, we have 4 X minus 4 X here. So that's 2 + 6, which is 8, all divided by 2 X + 1 all squared multiplied by uh 2 X + 1 squared multiplied by 2 X + 1 squared, which is 2 X + 1 to the 4th. And this is gonna be 24 multiplied by 2X minus 3 squared divided by 2 X plus 1 to the 4th, OK? So just to recap, we found our derivative of our other function. We use the quotient rule to find the derivative of the inner function, and then we plugged it back into our problem to get the derivative of F of X. Now that we have the derivative of F of X, we can find F of G of negative 2. So now, that means we're just gonna substitute G of negative 2. Into the derivative of F of X. And remember earlier we said that it is 0. So now, this is gonna be 24 multiplied by 2 multiplied by 0 minus 3 squared, OK, divided by 2 multiplied by 0 plus 1 to the 4th. And now that's gonna be 24 multiplied by -3 squared, OK, which is 9. Divided by 2 multiplied by 0 is 0, so that's gonna be 1 to the 4th or 1, and the 24 multiplied by 9 is 216. OK. So F of G of negative2 equals 216. And since that's the case, if we go back to our original problem because now we have all the information we need, F of G of -2, OK, then is gonna be equal to F G -2, 216 multiplied by G of -2, which we found to be 1, and 216 multiplied by 1 equals 216. Thus, our answer is 216, which means C is the correct option. Thanks a lot for watching everyone. I hope this video helped.

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

Key Concepts

Chain Rule

Composite Functions

Derivative of Rational Functions

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