Why Didn't the First Method Work for Finding the Derivative of y = x^(e^x)?

[rainyrabbit]

Finding derivative question please...

How should I find the derivative of y = x^(e^x)?

I tried using the chain rule along with the power rule, coming out to:
(e^x) (e^x) (X^(e^x - 1))

If I had took the natural log of both sides and then used implicit differentiation, I would have gotten as a derivative:
(x^(e^x)) (e^x) (1/x + ln x)
which is the correct answer according to my TI89.

Why wouldn't the first method work? Or was there any flaw?

By the way I just started Calculus as a high schooler.

 

[courtrigrad]

The power rule only works for functions of the form $$f(x) = x^{n}$$. So you can't use it for functions in the form of $$f(x) = x^{g(x)}$$ You could use implicit differentiation. Or you could do the following:

$$x^{e^{x}} = (e^{\ln x})^{e^{x}$$.

$$\frac{d}{dx} ( x^{e^{x}})= \frac{d}{dx}(e^{\ln x}^{e^{x}}) = e^{\ln xe^{x}}\frac{d}{dx}(e^{x}\ln x)$$

$$\frac{dy}{dx} = e^{\ln x e^{x}}(\frac{e^{x}}{x}+e^{x}\ln x)$$

$$\frac{dy}{dx} = x^{e^{x}}e^{x}(\frac{1}{x}+ \ln x)$$

Note: $$e^{\ln x e^{x}} = (e^{\ln x})^{e^{x}} = x^{e^{x}}$$