How can the derivative of x^x be solved?

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Discussion Overview

The discussion centers on finding the derivative of the function x^x, exploring various methods and approaches to solve it. Participants share their understanding of calculus concepts relevant to this problem.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to solve the derivative of x^x, expressing uncertainty about the formulas applicable to the problem.
  • Another participant suggests taking the natural logarithm of both sides, differentiating using the chain and product rules, and arrives at a formula for the derivative.
  • A different approach is proposed, indicating that x^x can be expressed as e^(x*ln(x)), suggesting differentiation from this perspective.
  • One participant humorously encourages learning the power rule but later acknowledges that it does not apply in this case since the exponent is not constant.
  • A later reply supports the logarithmic differentiation method as correct, challenging the applicability of the power rule in this context.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the power rule and the methods for finding the derivative, indicating that there is no consensus on a single approach.

Contextual Notes

Some participants rely on specific calculus techniques, such as logarithmic differentiation and the power rule, without fully resolving the applicability of these methods to the problem at hand.

The_Brain
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How do you solve the derivative of x^x? I'm sure it's fairly easy-- I'm just beginning calc though and none of the "forumlas" work.
 
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Take the natural log of both sides.
y=xx
lny=xlnx
Differentiate with respect to x.
(1/y)(dy/dx)=lnx+x/x (chain rule, product rule)
dy/dx=y(lnx+1)

You could do the same thing with y=xa, where a is a constant.
lny=alnx
dy/dx=y(a/x)=xa(a/x)=axa-1
a formula I'm sure you know well
 
Last edited:
or xx = ex*ln(x) ad continue from here...
(ex*ln(x))' = ex*ln(x) * (x*ln(x))' and so on...
 
Originally posted by The_Brain
How do you solve the derivative of x^x? I'm sure it's fairly easy-- I'm just beginning calc though and none of the "forumlas" work.
check out the power rule, learn it , know it, LOVE IT! Its easy, i am just playing but check it out, ok. Any problems with it let me know, k.
Dx :wink:
 


Originally posted by Dx
check out the power rule, learn it , know it, LOVE IT! Its easy, i am just playing but check it out, ok. Any problems with it let me know, k.


No, the power rule is not applicable here because the exponent is not a constant. Stephen Privitera's solution is correct; go with that.
 

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