Question

Find derivative of (tanx/x) log(e^x/x^x)?

Answer 1

`sec^2 x (1- log x) - tan x / x`

Explanation:

Let's start with simplifying the latter factor of the given expression a little:

# log ( e^x / x^x ) = log ( ( e / x )^x ) = x log ( e / x ) = x ( log e - log x ) = x ( 1 - log x ). #

Therefore, we have to find the derivative of

# tan x ( 1 - log x ). #

To that end, we use the product rule, and arrive at the final result

# sec^2 x ( 1 - log x ) - tan x / x. #