Which one is greater using Mathjax

Almost Everyone know the value of π(=) and e(=). but when we consider ,which one is greater , "π" ,absolutely easy !!!
But the question is which is greater π^e or e^π.
Not too difficult!!!
π^e equals and e^π equals .
But doing this, without calculator becomes difficult.
so, Let's crack the question with using some Maths.
Let's consider the function, $$f(x)=x^{x}$$Taking Log(base e) both sides,$$ ln (f(x)) = {\frac {1}{x}}ln(x)$$Differentiating both sides ,$${\frac {1}{f(x)}}f'(x) = -{\frac {1}{x^{{2}}}}ln(x) + {\frac {1}{x^{2}}}$$
After rearranging the terms ,$$ f'(x) = {x^{\frac{1}{x}}(1-ln(x)) \over x^{2}}$$ (I hope u understand !!)
Since , ln(x) is greater than 1 for x> e ;
It implies that f'(x) is always negative. Since ,as we know π >e , f(π) < f(e),$$\to \pi ^{\frac {1}{\pi}} < e^{\frac{1}{e}}$$ $$ \to \pi^{e} < e^{\pi}$$
Hence proved ,showed !!!!

Which one is bigger using Mathjax