Question

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

Answer 1

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

first of all, taking log both sides,

logy = x^x. loge

logy = x^x

now differentiate both sides with respect to x,

1/y . dy/dx = d(x^x)/dx = d[e^{xlogx}]/dx

1/y . dy/dx = e^{xlogx}[ x. d(logx)/dx + logx . dx/dx ]

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

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

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

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