I promise guys, no homework here, just curiosity.(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to find dy/dx for the equation y = x^x - C, where C is any arbitrary constant. I've found two ways that SHOULD be ok to take this derivative, but they produce different answers, I was wondering which method is correct and which method is incorrect. Also, why? It seems to me that both of these methods should be ok. Anyways, here they are:

Method 1:

[itex]y = x^x - C[/itex]

[itex]ln(y) = ln(x^x - C)[/itex]

[itex]ln(y) = \frac{x ln(x)}{ln(C)}[/itex]

Now take the derivative:

[itex]\frac{\frac{dy}{dx}}{y}=\frac{1}{ln(C)}(x ln(x))'[/itex]

Using the Product Rule, it can be seen that [itex](x ln(x))' = ln(x) + 1[/itex]. Therefore:

[itex]\frac{dy}{dx}=\frac{y}{ln(C)}(ln(x) + 1)[/itex]

[itex]\frac{dy}{dx}=\frac{x^x}{ln(C)}(ln(x) + 1)[/itex]

Method 2:

[itex]y = x^x - C[/itex]

[itex]y' = (x^x)' - C'[/itex]

[itex]y' = (x^x)'[/itex]

[itex](x^x)'[/itex] can be evaluated using method 1 for the equation [itex]y = x^x[/itex]

[itex]\frac{dy}{dx} = x^x(ln(x) + 1)[/itex]

Method one seems a bit less hand wavy, so I'm more confident in it; however, the derivative shouldn't depend on C, so that makes me lean more toward Method 2.

Anybody have any input they'd be willing to share?

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# How to take this derivative?

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