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Tutorial ExorciseFind the derivative of the logarithmic function. ((x) = In(x57...

Question

Tutorial ExorciseFind the derivative of the logarithmic function. ((x) = In(x57

Tutorial Exorcise Find the derivative of the logarithmic function. ((x) = In(x57



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Use logarithmic differentiation to find the derivative of the function. $$y=x^{\ln x}$$

In a lot of math applications, we use exponential functions. Um, and because exponential functions are extremely useful in math applications, especially in calculus, we want to be ableto take derivatives. We want to find the rates of change of those specific exponential function. So in this case, what we have is we have a Y equal to X to the X, and we ultimately want to take the derivative of this. So one thing we can do it first. To simplify things is take the natural log of both sides. This will allow you to perform implicit differentiation which will ultimately and becoming easier in the long run. Now we can differentiate both sides. So when we do that, we differentiate this side right here. What will end up getting is that one over? Why times wide crime, of course, is equal to one over. Ah x times X class. Another thing we keep minus. When this is raised to the power of an air component, we can get rid of that and put it up here. So we're doing now. Is the product rule? Yes, we have one over X times acts plus the natural log of X times The derivative of executions can be one we see that X times one over access just one. So now we have that one over. Why times why Prime is equal to one. Plus the natural ago vax well, multiply. Why on both sides to get this right here, Then we know that why is going to be X to the X? Because that was already defined as our original problem. So I'm just using that substitution and solving for white crime through implicit differentiation. We see that why prime is equal thio x to the X times one plus the natural log of X. And this is one helpful way that we can, um performed derivatives on exponential functions in the future.

This question asked us to use longer mint lager with Mick differentiation to find the derivative of the function. First things first. Take the natural with both sides. Now differentiate both sides with respect to X and we have y prime is axe times. The natural log of X is one over X plus the natural log of acts times one which gives us why prime is acts the power of acts times one plus natural log of X. This is also really common. Problem in calculus is why is extra power axe? You can understand just using the normal ideas of natural log and taking the natural log of both sides and then Ellen of X is one over.

F. F. X. Equal to X. Time natural O. X. So here in this kitchen we have to find the Lord ethnic derivative. So here we know that for the Lord ethnic derivative. B. Natural law F. Of X upon D. X. Equal to derivative of F. Of X. Upon fo facts. So now we apply the property here and we get D. Natural Law fo facts upon the X. So here we know that their derivative of F. Of X. So we can write like that. X. Time Natural X upon X. Natural low X. So first of all we have to find the derivative all X. Time Natural X. So here we get one time one upon X. Sorry X. Time one upon X. Blood. One time natural O. X. So apply after playing product rule for delivery. You know, we simplify this and we get one plus natural log X. So now we have to put that value here and we get one plus natural log eggs A bone X. Time Natural act. So it is our final answer.

At four X. Equal to natural Oh, Act. So here we have to find the logarithmic derivative. And here we know that for the lloret make derivative D natural low F of X upon D X. Equal to derivative of F of X upon F of X. So now we apply that property here and we get D Natural law. And for facts upon D. X equal to derivative of natural X upon natural law X. So here you can see that the derivative of natural X, which is one upon X. So now now we put that value here, so we get one upon X. Time. Natural look X. So it is all final answer.


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