Find f'(7) for f(x)=6x^ln(x)

Answer 1

Use chain rule.
Let
y=6x^ln(x)
take log on both sides:
ln(y) = ln(6)+ln(x)*ln(x)
ln(y) = ln(6) + ln²(x)
differentiate both sides w.r.t. x
(1/y)y' = 0 + 2ln(x)*(1/x)
y' = dy/dx
= y*2ln(x)*(1/x)
= 6x^ln(x)*2ln(x)*(1/x)
= 12x^ln(x)-1*ln(x)

Check my calculations.