75–86. Logarithmic differentiation Use logarithmic differentia...

Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (x+1)¹⁰ / (2x-4)⁸

Accepted Answer

Welcome back, everyone. In this problem, we want to use logarithmic differentiation to find the derivative of F of X equals 3 x + 2 5th divided by X minus 3 to 7th. A says it's 3 x + 2 cubed multiplied by 15 X minus 222, divided by X minus 3 to 14th. B 3 X + 2 to 4th multiplied by 6 X + 59 divided by X minus 3 to 8th. C-3 X + 25th multiplied by 7 X minus 43 divided by X + 3 to 9th and the D 3 X + 2 to 6th multiplied by 2 X + 3 divided by X minus 3 to 12. Now, if we're going to use logarithmic differentiation to find the derivative of F of X, then first, let's try to rewrite our term in logarithmic format so that we can use that type of differentiation, OK? Now, let's start by letting Y be equal to FFX, OK. Which means that we can rewrite our problem as Y equals 3 x + 25th. Divided by x minus 3 to the 7th. Now we can take the logs off both sides, OK? And when we do that. Then we're left with the natural log of Y being equal to the natural log of 3 X + 2 to 5th. Divided by x minus 3 to 7th. Now what's helpful about writing it in this format? Then we can use the properties of logarithms. Recall, OK, that the natural log of A divided by B is equal to LNA minus LNB. So if we apply that property of logarithms here, OK, then this means. That we could rewrite our problem as LNY equal LN 3 X + 2 rates to the 5th. Minus L N X minus 3 raised to the 7th. We can take it a step further by applying another property of logs. Well that property also tells us that if we're finding the log of a term race or a power, we can bring down the power. In other words, LNA to the X equals XLNA. So if we use that property, then we could rewrite our problem even further as 5 LN 3 x + 2. We bring down the 5. 7 LN X minus 3. No, that makes it a little easier for us because we won't have to use the chain rule, uh, to differentiate anything with to a power. Now that we have our problem written with logs, then we can go ahead and differentiate both sides of the given equation, OK? And know by differentiating. With respect to X, we'll have to differentiate implicitly for LN Y, OK. And when we do that, then we're gonna get 1 divided by Y, multiplied by the derivative of Y equal to 5 multiplied by the derivative of LN 3 X + 2. -7 multiplied by the derivative of LNX minus 3, OK? Now we can differentiate, OK, because remember that when we differentiate our logarithmic term. OK, differentiate a logarithmic term. We also know, let me put that at the side here. That The derivative of a natural log function equals one divided by the function multiplied by the derivative of the function, OK? So in that case, if we apply that here. And then this is gonna become 5 multiplied by 1. Divided by 3 x + 2. Multiplied by the derivative of 3 x + 2, which is 3, minus 7 multiplied by 1 divided by X minus 3 multiplied by the derivative of X minus 3, which is 1. The derivative of X is 1, the derivative of minus 3 is 0. So now when we simplify that we're left with 15 divided by 3 X + 2. 7 divided by X minus 3. Now we can take it a step further to solve for the derivative of Y, OK? We can multiply both sides of our equation by Y. And now when we do that, let me write back the 3 here. When we do that. Then we're left with Y. OK, sorry, let me just clean this up again. We're left with Y being equal to Y multiplied by our expression, so that's Y multiplied by. 15 divided by 3 X + 2 minus 7 divided by X minus 3, OK. But recall, we know that Y is F of X, so we can replace Y with F of X in our problem to get our expression purely in terms of X. So that's gonna be 3 X + 2 res to the 5th divided by X minus 3 res to the 7th. And no, for expression, we can make it one fraction, OK, which is gonna be 15 multiplied by X minus 3, minus 7 multiplied by 3 X + 2, all divided by 3 X + 2. Multiplied by x minus 3. And we can do some simplification here, OK? For starters, we know that 3 X + 2, we're going into 3 X + 2 to the 5th and leave it with 3 X + 2 to the 4th, OK? Likewise, X minus 3 to the 7th multiplied by X minus 3 is gonna leave us with X minus 3 to the 8th. And if we expand the terms in our numerator here, then our problem is now going to become. 3 X + 2 to the 4th. Divided by X minus 3 to the 8th multiplied by 15 X minus 45 minus 21 X minus 14. That is when we expand our brackets and now if we simplify that, OK, that's 3 X + 2 to the 4th divided by X minus 3 to the 8th. Now that's gonna be multiplied by. 15 X. Well, here we have, sorry, 15 X minus 21 X. That's 6 X. 45 minus 14, that's 59. And if we factor out our negative sign, we're gonna be left with -3 X + 2 raise to the fourth divided by X minus 3 raises to the 8th multiplied by 6 X plus 59, OK? So this is going to be the derivative of Y. In other words, the derivative of F of X. And let me just write it in the middle, just to make it look a little bit better, OK? So now, by logarithmic differentiation, this is the derivative of FFX. If we go back to our answer choices, that tells us that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)¹⁰ / (2x-4)⁸

Key Concepts

Logarithmic Differentiation

Product and Quotient Rules

Chain Rule

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