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Problem 36 Hard Difficulty

Find $ f'(a) $.

$ f(x) = \dfrac{4}{\sqrt{1 - x}} $

Answer

$f^{\prime}(a)=\frac{2}{(1-a)^{\frac{3}{2}}}$

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Video Transcript

here we have the function F. Of x equals four over the square root of one minus X. We want to find the derivative F. Prime of X. Ah The first thing I wanna do is I want to rewrite this expression as for times one minus X To the negative 1/2 power. Let me explain one minus X. To the negative one half power. One minus X. To the positive one half power would be the square root of one minus x. One minus X. To the negative one half power would be one over the square root of one minus sex. And that's exactly what we have here because uh square root of one minus X. And the denominator is really the same thing as one over the square root of one minus sex. So another way you can think of it is uh the square root of one minus X is really one minus x to the one half. And when I moved that radical Uh to the opposite side of the fraction. So from the bottom of the fraction to the top 1 -1 to the positive one half. When written up here becomes one minus X. To the negative one half. And that's how we have four times one minus x. To the negative one half. Now define F prime of X. We are going to have to take the derivative with respect to X. Of this expression for Times one -X. To the negative 1/2 power. So we had to take the derivative of this function. Now we're going to have to use the chain rule. And the chain rule says if we have some function you ah if we have some function you of X. Let me erase and rewrite it. If we have a function of facts, he was a function of X. Then the derivative of F. With respect to you. I'm sorry. With respect to X equals the derivative of F. With respect to you times the derivative of you with respect to X. So, if F is a function of you, if we could write F as a function of you and in turn you is a function of X than the derivative of F with respect to X will equal the derivative F. With respect to you. First times times the derivative of U. With respect to X. All right, let me show you how this is gonna work. You is going to be one -X. So, what we really have here is F. Of you is equal to four Times 1 -1. Which is our you To the negative 1/2 power. So, we do want to find F. Prime of X. Which is the derivative by our function F with respect to X. But we need to use to change rules. I let you be the one minus X. So, we rewrote our function F of X. Which was the four times one minus X. To the negative one half. We rewrote it in terms of you. Since this is U. F. Of you is really four times U. To the negative one half? That's exactly what we have here F. Of U equals four times U. To the negative one half. Uh huh. No, since f of U equals uh let me show, make sure we still see the U function. We're going to need it. And our chain rule since F. Of U equals four times you. Two negative one half. Of course you being this uh define F prime of X. D. F. D. X. Okay. DF dx which is our F prime of X. Okay. Is going to equal All right? We're applying uh We are applying this chain rule now. Yeah. F prime of X. DF dx Same thing as F prime of X. Is equal to the derivative of our function F. With respect to you. Well, here's our function F of you. What is the derivative of this function? Uh With respect to you? Well, we're gonna have the four And derivative of U. to the negative one half. With respect to you. The derivative of U to the negative 1/2 Is negative 1/2 times you. And then uh you have to subtract the one from this exponent. So negative three hits. So this the D. F D. U derivative of F of you is the derivative of this function with respect to you which comes out to be four times the derivative of you which is negative one half times you. Two negative three years. Now. To finish the chain rule, we have to multiply this by D. U. D. X. Okay. What is the derivative of U. With respect to X. While our you of X. Function U. Is equal to one minus X. So the derivative of U. With respect to X. Is the derivative of one minus X. What is the derivative one minus X? Uh While the derivative of one is zero, the derivative of negative X. Negative one. X. Is negative one. So D. U. D. X. Derivative of our you function with respect to X. Is negative one. So D. E. F. D. X. I'm gonna rewrite it as F. Prime of X. That's what DF dx means. DF dx and uh F. Prime of X. Derivative of F. With respect to X. And F. Prime of X. These are the same thing. They mean the same thing. Uh this is equal to uh well let's 2-4 times negative one. That's negative for. Um And actually let's let's multiply it by the negative one half also. So four times negative two. I'm gonna have to be racist negative four times negative one half is negative two negative two times negative one is positive two. So we have to times U. To the negative three hips. But we want to rewrite this as a function with just X. No more. You. That's all we do is take what you was equal to and plug it back in? Once again four times negative one half times negative one is two now replacing for you. Ah The one minus X gets plugged back in now for U. U. Was equal to one minus X. So instead of you, we are going to write one minus X. And that was raised to the -3 # Power. And we are done as long as said, let me put a nice little box around it. Okay, so F prime of X comes out to be two times one minus X to the negative three hash power. So if you want to find F prime of a uh that is simply two times one minus A. To the negative three has power.