Thread: Derivative of this exponential function?

1. Derivative of this exponential function?

what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

2. Originally Posted by mneox
what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

differentiate x&#x5e;&#x28;e&#x5e;x&#x29; - Wolfram|Alpha

3. Originally Posted by mneox
what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

Write $f(x):=x^{e^x}=e^{e^x\ln x}\Longrightarrow f'(x)=e^{e^x\ln x}\left[e^x\ln x+\frac{e^x}{x}\right]$ $=x^{e^x}\left[e^x\ln x+\frac{e^x}{x}\right]$ $=x^{e^x}\ln\left(x^{e^x}\right)+e^xx^{e^x-1}$ .

Tonio