#4. What is the derivative of y = x^(ln x) where ln x is the natural logarithm of x.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `y = x^(ln x)` . The derivative `dy/dx` has to be determined.

First, take the natural log of both the sides. As `y = x^(ln x)` , `ln y = ln(x^(ln x))`

Use the logarithmic formula: `ln a^b = b*ln a`

`ln y = ln(x^(ln x))`

=> `ln y = ln x*ln x = (ln x)^2`

Now, using implicit differentiation and the chain rule, `(1/y)(dy/dx) = 2*ln x*(1/x)`

`dy/dx = y*2*ln x*(1/x)`

Substituting `y = x^(ln x) ` gives `dy/dx` = `2*ln x*x^(ln x)*(1/x)` = `2*ln x*x^(ln x - 1)`

The required derivative of `y = x^(ln x)` is `dy/dx = 2*ln x*x^(ln x - 1)`

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