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Question
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)`
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