What is the derivative of y = x^(x^x)?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to differentiate y = x^(x^x)

y = x^(x^x)

ln y = ln[x^(x^x)]

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

take the derivative of both the sides with respect to x

=> (1/y)*dy/dx = (x^x)/x + (ln x)*(x^x)'

let z = x^x, ln z = x*ln x

1/z(dz/dx) = x/x + ln x

=> dz/dx = x^x + x^x*ln x

=> (1/y)*dy/dx = (x^x)/x + (ln x)*(x^x*ln x+x^x)

=> dy/dx = y[(x^x)/x + (ln x)*(x^x*ln x + x^x)]

=> dy/dx = x^(x^x)[(x^x)/x + (ln x)*(x^x*ln x + x^x)]

The required derivative is dy/dx = x^(x^x)[(x^x)/x + (ln x)*(x^x*ln x + x^x)]

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