Question

Find the derivative of x^{log x} w.r.t. log x.

Answer 1

Let u = x^logx, v = logx and u = x^logx

Taking log on both sides

logu = logx^logx

logu = logxlogx

logu = (logx)²

Differentiate w.r.t. x

`1/u (du)/(dx) = 2 log x xx 1/x`

`(du)/(dx) = u . (2 log x)/x`

`(du)/(dx) = (x^(log x).2 log x)/x`   ...(i)

v = log x

Differentiate w.r.t. to x

`(dv)/(dx) = 1/x`   ...(ii)

On dividing equation (i) by equation (ii)

`((du)/(dx))/((dv)/(dx)) = (x^(log x) * (2 log x)/x)/(1/x)`

`(du)/(dv) = 2 logx.x^(logx)`

The derivative of x^logx w.r.t. to logx is 2logx.x^logx.