Calculus - Differentiating Logarithmic Functions

Question

HELP

Find the derivative of the function f(x) = x²(sinx)^x using logarithmic differentiation.

A) x²(sinx)^x(2x + ln(sinx) + xsin(x)xcot(x)

B) (2)/(x) + ln(sinx) + xcos(x)

C) 2x(sinx)^x + x³(sinx)^x-1

D) x²(sinx)^x(2)/(x) + ln(sinx) + xcot(x)

Answer 1

y = x^2 ( sinx)^x

ln y = ln [x^2 (sinx)^x]

ln y = ln x^2 + x ln sin x

y'/y = 2x/x^2 + x cosx/sinx+ ln sinx

= 2x/x^2 + x cotx + ln sinx

= 2/x + x cot x + ln sinx

then y = [2/x + x cot x + ln sinx]* y

= [2/x + x cot x + ln sinx] * x^2 (sinx ^x)