What is the value of f'(6) if f(x) = 4x^(ln(x))

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We have f(x) = 4x^(ln(x)) and we need to determine f'(6).

Let y = f(x) = 4x^(ln(x))

take the logarithm of both the sides

ln y = ln [ 4x^(ln(x))]

use log(a*b) = log + log b

=> ln y = ln 4 + ln(x^(ln x)

=> ln y = ln 4 + ln x*ln x

=> ln y = ln 4 + (ln x)^2

differentiate both the sides

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

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

f'(x) = 4x^(ln(x))*(2/x)(ln x)

f'(6) = 4*6^(ln 6)*(2/6)(ln 6)

=> (4/3)*(ln 6)*6^(ln 6)

The required value of f'(6) = (4/3)*(ln 6)*6^(ln 6)

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