dy/dx=x^x^x

Question

differentiate dy/dx =  x^x^x

Answer 1

differentiate dy/dx =  x^x^x

 

Let u = (x^x)^x.

Let v = x^x.

Then u = v^x,

Also, ln(v) = x.ln(x).

Differentiating both sides wrt x,

(1/v).dv/dx = ln(x) + x/x

dv/dx = (x^x)(1 + ln(x))

From u = v^x,

ln(u) = x.ln(v)

Differentiating both sides wrt x,

(1/u).du/dx = ln(v) + x.(1/v).dv/dx

du/dx = u{ln(v) + (x/v).(x^x).(1 + ln(x))

du/dx = ((x^x)^x){ln(x^x) + x(1 + ln(x))}