Differentiating y=x^x: Acceptable Method?

In summary, the conversation discusses differentiating the function y=x^x with respect to x=ln(y). The base is then changed to e, resulting in x=ln(y)/ln(x). The calculator gives an answer of x^{x(1+ln(x))} and the group notices that this can be simplified to x^x. There is some discussion about whether the factor 1+ln(x) should be in the exponent or not. The conclusion is that the correct answer is x^x(1+ln(x)).
  • #1
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Differentiating y=x^x

x=ln(y)

I changed the base to e

[tex]x=\frac{ln(y)}{ln(x)}[/tex]
[tex]xln(x) = ln(y)[/tex]
[tex]e^{xln(x)} = y[/tex]
[tex]e^{xln(x)}(1+ln(x) = \frac{dy}{dx}[/tex]

The answer the calculator got was [tex]x^{x(1+ln(x))}[/tex] so I noticed that since [tex]y=x^x[/tex] and [tex]e^{xln(x)} = y[/tex], then I could replace it with x^x in the final answer

Is this an acceptable method? Is there any circular logic I missed? Could I leave it as is wihtout writing x^x?
 
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  • #2
yes, but the factor [itex]1+\ln x[/itex] should not be in the exponent (chain rule).
 
  • #3
^ Thank you...yeah I made an error there
 

Related to Differentiating y=x^x: Acceptable Method?

What is the acceptable method for differentiating y=x^x?

The acceptable method for differentiating y=x^x is to use the logarithmic differentiation technique.

What is logarithmic differentiation?

Logarithmic differentiation is a technique used to find the derivative of a function that is in the form of y=x^x. It involves taking the natural logarithm of both sides of the equation and then using the rules of logarithms to simplify the expression.

Why is logarithmic differentiation used for differentiating y=x^x?

Logarithmic differentiation is used for differentiating y=x^x because it is a more efficient and accurate method compared to the power rule or product rule. It also allows for differentiation of more complex functions, such as y=(x^2)^x.

What are the steps for using logarithmic differentiation to find the derivative of y=x^x?

The steps for using logarithmic differentiation to find the derivative of y=x^x are:
1. Take the natural logarithm of both sides of the equation
2. Use the rules of logarithms to simplify the expression
3. Differentiate both sides of the equation using the chain rule
4. Solve for y' to get the derivative of y=x^x

Are there any limitations to using logarithmic differentiation for differentiating y=x^x?

Yes, there are some limitations to using logarithmic differentiation for differentiating y=x^x. It can only be used for functions in the form of y=x^x and cannot be applied to other types of equations. Additionally, it can be more time-consuming compared to other methods for simpler functions.

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