Use logarithmic differentiation to find the derivative of the function.

Question

y = (ln(x))^cos(3x)

Answer 1

Take the logarithm of both sides to get:

ln y = cos(3x)ln(ln(3x))

Taking the derivative of both sides, we have:

y'/y = -3sin(3x)ln(ln(3x)) + cos(3x)/xln(3x)

Therefore, since y = (ln(x))^cos(3x), we have:

y' = y(cos(3x)/xln(3x) - 3sin(3x)ln(ln(3x))) = (ln(x))^cos(3x)(cos(3x)/xln(3x) - 3sin(3x)ln(ln(3x)))