Use logarithmic differentiation to find the derivative of the function.

Question

y = (tan(x))^7/x

Answer 1

Taking the logarithm of both sides, we have:

ln y = 7ln(tan x)/x

Taking the derivative of both sides, we have:

y'/y = 7(-ln(tan(x))/x² + sec²x/(x tan x))

Since y = (tan(x))^7/x, we have:

y' = 7y(sec²x/(x tan x) - ln(tan(x))/x²) = 7(tan(x))^7/x(sec²x/(x tan x) - ln(tan(x))/x²)