My question is on differentiation

Question

Can anyone differentiate this without logarithmic differentiation?

(log x)^{cos x}

Answer 1

desmos.com/calculator/ijelol40ml

Answer 2

derivative of (lnx)^cosx = [(lnx)^cosx](cosx/ln(lnx) - sinxlnx(lnx)]

if you were taking a beginning course with an introduction to logs, then "log" woud be the common log with base 10. but this problem is suffiently advanced mathematics that "log" would have to mean the natural log with base e, "ln"

derivative of lnx^cosx

= (lnx^cosx)((cosx)'ln(lnx) + cosx(ln(lnx)')

= lnx^cos(1-sinx)(ln(lnx0 + cosx/lnx)(lnx)')

= lnx^cosx(cox/x/ln - xinln(lnx)

= lnx^cos(cosxlnx - sinln(lnx)