Question

Differentiate the following function with respect to x: `(log x)^x+x^(logx)`

Answer 1

`Let y=(logx)^x+x^(logx).............(1)`

`Now `

`y=y_1+y_2 ..........................(2)`

Differentiating (2) with respect x, we get 

`dy/dx=dy_1/dx+dy_2/dx.........(3)`

Now take log of y₁ = (log x)^x

`log y_1 = x log (log x)`

Differentiating with respect to x, we get

`1/y_2 dy_2/dx=(2logx) xx 1/x`

`dy_2/dx=y_2((2logx)/x)=x^(logx)((2logx)/x)................(5)`

Adding equation (4) and (5), we get:

`dy/dx=(logx)^x(1/logx+log(logx))+x^(logx)((2logx)/x)`