Question

Differentiate x^x with respect to x log x.

Answer 1

Step 1: Assign variables to the functions

u = x^x

v = x log x

Taking log on both sides, we will get

We need to find `(du)/(dv)`, which is given by,

`(du)/(dv) = ((du)/(dx))/((dv)/(dx))`

Step 2: Differentiate u with respect to x

u = x^x, use logarithmic differentiation:

log u = x log x

Differentiating both sides with respect to x:

`1/u . (du)/(dx) = x . 1/x + log x . 1 = 1 + log x`

`(du)/(dx) = u(1 + log x) = x^x(1 + log x)`

Step 3: Differentiate v with respect to x

v = x log x use the product rule:

`(dv)/(dx) = x . 1/x + log x. 1 = 1 + log x`

Step 4: Divide the results

`(du)/(dv) = (x^x(1 + log x))/(1 + log x)`

= x^x