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Problem 43 Easy Difficulty

Use logarithmic differentiation to find the derivative of the function.

$ y = x^x $

Answer

$$y^{\prime}=x^{x}(1+\ln x)$$

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Video Transcript

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.

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