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

Question

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

f(x) = x^10x

Accepted Answer

Welcome back, everyone. In this problem, we want to use logarithmic differentiation to find the derivative of F X E equals 2 X-rays to the power of 5 X. A says it's 2 X to 4X multiplied by LN2X + 1. B 5X multiplied by 2 X 5X minus 1 multiplied by LN2X + 1. C, 2 X-rays to 5 X multiplied by LN2X + 1. And D 5 multiplied by 2 X to 5 X multiplied by LN 2 X plus 1. Now, before we use logarithmic differentiation to find our derivative derivative, let's try to write it in a simpler way, OK? Now, here, we're raising one term to the other, and both of them have variables, OK? And given. F of X being equal to 2 X raised to the power of 5 X. OK. Then we can take the natural logarithm of both sides to bring our power down. Because no, that means then. And if we let Y be equal to F of X, OK. Then we're gonna have Y being equal to 2X raised to the power of 5 X, which means. Let me fix that here. Which means that by taking the natural log of both sides, the natural log of Y will be equal to 5. Well, let me write it out first. The natural log of 2 X rays to the power of 5 X. And now using what we know about the poor rule of logarithms, we can bring ourpo down to get the natural log of Y. To be equal to 5 X multiplied by the natural log of 2 X. Now here, notice that we have Y on this side, um, the natural log of Y on the left hand side, not just Y. So we'll need to differentiate implicitly, OK? So we can differentiate. OK. To solve for the derivative of way. Because now when we do that. Then on our left hand side, we're going to have the derivative of the natural log of y, and on the right hand side we're gonna have the derivative, OK, with respect to X of 5 X multiplied by the natural log of 2 X. Now, let's differentiate both sides. OK, so now on the left hand side. OK, then the derivative of the natural log of Y will be equal to. 1 divided by Y multiplied by y. Which equals y divided by y. So we have an expression here. Now hobo on our right hand side, the derivative of 5 X multiplied by the natural log of 2 X. Well, here OK. Uh, here the derivative. So now this is on the right hand side. Notice that this is a product, OK. So we're finding the derivative of 5 X multiplied by the natural log of 2 X. So we can apply the product tool of differentiation. So we're gonna write down the derivative of 5 X, which is just going to be 5, and we're gonna multiply it by our term, the other term which is LN2X. And now, on the other hand, OK. We're going to add that to the derivative of LN2X which would be 1 divided by X multiplied by 5 X. OK? And now when we do that, OK, we're gonna be left with 5 LN 2X. Plus 5 because X cancers X. Know that we have the derivative of both sides, then if we put them into our problem, then we're gonna have Y divided by Y being equal to 5. LN 2 X plus 5 or 5, LN 2 X plus 1. If we factor out 5. This means then that the derivative of Y. Is going to be equal to 5 Y multiplied by LN 2X plus 1 and know if we substitute Y for what we know it was remember Y was equal to F of X then we could rewrite this as 5 multiplied by 2 X raised to the power of 5 X multiplied by the natural log of 2 X plus 1. Thus, this is the derivative of Y or FF X. If we look back on our answer choices, that tells us then that D 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^10x

Key Concepts

Logarithmic Differentiation

Chain Rule

Product Rule

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