Question

Use logarithmic differentiation to find the derivative of the function.

y=(sin(7x))^ln(x)

y'=? ...?

Answer 1

`y=(\sin(7x))^(\ln(x) ) \\ \\ ln(y)=\ln[(\sin(7x))^(\ln(x) )] \\ \\ ln(y)=\ln(x)\ln[(\sin(7x))] \\ \\ (ln(y))'=(\ln(x)\ln[(\sin(7x))])' \\ \\ (1)/(y) dy=[\frac{1}x}\ ln \sin(7x) +(7 \cos (7x))/(sin(7x))ln(x)] dx \\ \\ (dy)/(dx) =y[\frac{1}x}\ ln \sin(7x) +(7 \cos (7x))/(sin(7x))ln(x)] \\ \\ (dy)/(dx) =(\sin(7x))^(\ln(x) )[\frac{1}x}\ ln \sin(7x) +(7 \cos (7x))/(sin(7x))ln(x)]`