Question

Differentiate the function w.r.t.x. : ${x^{e}}^{x}$

Answer 1

we have to differentiate $y=x^{e^x}$

first of all, take log both sides,

logy = e^x.logx

now, differentiate both sides with respect to x,

1/y . dy/dx = logx. d(e^x)/dx + (e^x). d(logx)/dx

1/y . dy/dx = logx . e^x + e^x . 1/x

1/y . dy/dx = e^x [ logx + 1/x ]

dy/dx = y. e^x [ logx + 1/x ]

dy/dx = x^{e^x}. e^x [ logx + 1/x ]