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Question
Differentiate \[x^{1/x}\] with respect to x.
Solution
Let `y=x^(1/x)`
Take the natural logarithm of both sides: `log y = 1/x log x`
Differentiating log y gives:
`1/y dy/dx = d/dx (log x/x)`
`d/dx (log x/x)=((1/x)xxx-logxxx1)/x^2`
`1/y dy/dx = (1-logx)/x^2`
Multiply through by y to isolate `dy/dx`
`dy/dx = yxx (1-logx)/x^2`
Substitute y = x1/x back into the equation:
`dy/dx = x^(1/x) xx (1-logx)/x^2`
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