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

Question

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

f(x) = (1+ 1/x)^x

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of the function F of X is equal to 3 + 1 divided by X, all to the power of 2 X using logarithic differentiation. Awesome. So it appears for this particular prompt, we're asked to take our function of FFX and we're asked to find what the derivative of this function of FFX is. And that's what we're ultimately trying to solve ours we're trying to figure out what the derivative of FFX is using logarithic differentiation. So with that said, first off, let us state that Y is going to be equal to 3. Plus 1 divided by X all to the power of 2 X. Let us say that Y equals our function that we are given by the problem itself. And then we need to take this expression for Y, and we need to take the natural logarithm of both sides to in order for us to recall and apply the power rule. So let me do that, we will find that the natural log of Y is equal to the natural log of 3 + 1 divided by X all to the power of 2 X. Fantastic. And when we do a little bit of simplifying, we will find that the natural log of Y is going to be equal to 2 X multiplied by the natural log of 3 + 1 divided by X, fantastic. And that was our first step. Now our second step is we need to differentiate the left and right hand sides, and for the left hand side, we're going to have to recall and apply the chain rule, since, as we should recall, let's make a note of this, that since Y is equal to Y of X, and because Y is equal to Y of X, we're going to have to recall and apply the chain rule to the left hand side. So, for the right hand side, on the other hand, we need to recall and apply the product rule of differentiation such that, and as we should recall, F multiplied by G. Of of X multiplied by X is going to be equal to F of X multiplied by G of X plus F of X multiplied by G. Of X. Fantastic. So with that in mind, we need to go ahead and write that 1 divided by Y multiplied by Y is equal to 2 multiplied by X multiplied by the natural log of 3 + X to the power of -1. Fantastic. And then we need to make sure that this is all to the power of prime or this prime or this like, and essentially when you have an apostrophe in the power, this prime symbol indicates you're taking the first derivative. Awesome. So then we could go ahead and do a little bit of simplifying and we could write that Y divided by Y is equal to 2 multiplied by X multiplied by the natural log of 3 + X to the power of -1. Plus X multiplied by the natural log prime of 3 plus X to the power of -1. Fantastic. So with that said, our next step. As we need to go ahead. For our third step, we need to go ahead and apply the chain rule for the logarithm at the end of the previous lines we need to take Y divided by Y is equal to 2 multiplied by the natural log of 3 + 1 divided by X plus X multiplied by 1 divided by 3 + 1 divided by X multiplied by 3 + X to the power of -1. Prime, fantastic. And when we do a little bit of simplifying, we will find that Y divided by Y is equal to 2 multiplied by the natural log of 3 + 1 divided by X plus, and then we need to take and we need to add X divided by 3 + 1 divided by X multiplied by negative X to the power of -2. Fantastic. So now, Our next step, so this is our step 4, is we need to rewrite our expression by cleaning it up a bit, and when we do, we will find that Y divided by Y is equal to 2 multiplied by the natural log of 3 + 1 divided by X minus X divided by 3 X + 1 divided by X multiplied by 1 divided by X2. Which is going to be all equal to 2 multiplied by the natural log of 3 + 1 divided by X, and then when we take 3 + 1 divided by X and then we're gonna need 2. Subtract. 1 divided by 3 X + 1. And then don't forget your final parenthesis. Fantastic. So, with that said, our next step now, this is step 5. So for our 5th step, We need to solve for Y by rearranging the equation that we found in step 4 and substituting in our value for Y. So when we do that, we will find that Y is equal to 3 + 1 divided by X, the power of 2 X multiplied by 2 multiplied by the natural log of 3 + 1 divided by X minus 1 divided by. 3 X + 1. Fantastic. And our 6th and final step, so this is our last step, we need to write everything under one fraction, so we need to write that Y is equal to 2 multiplied by 3 X plus 1, divided by X all to the power of 2 X multiplied by the natural log of 3 + 1 divided by X minus 1 divided by. 3 X + 1, fantastic. So when we do a little bit of simplifying, we will find that Y Is going to be equal to drumroll, 2 multiplied by 3 + 1 divided by X to the power of 2 X multiplied by 3 x + 1 multiplied by the natural log of 3 + 1 divided by X minus 1, all divided by 3 X + 1. And that's it. That is our final answer. Hooray, we did it. So, once again, in conclusion, the derivative of the function F of X, so the derivative of F of X is going to be Y is equal to 2 multiplied by 3 + 1 divided by X to the power of 2 X, all multiplied by 3 x + 1 multiplied by the natural log of 3 + 1 divided by X, minus 1 all divided by 3 X plus 1. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+ 1/x)^x

Key Concepts

Logarithmic Differentiation

Chain Rule

Exponential Functions

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