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Use logarithmic differentiation to find the derivative of the function.

$ y = x^x $

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

00:38

Frank Lin

00:46

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

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Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

01:44

Use logarithmic differenti…

02:08

01:49

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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