# Differentiate the following function with respect to x: (log x)x+x(logx) - Mathematics

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

#### Solution

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)

Concept: Logarithmic Differentiation
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