Question

find the derivate of f(x)= (1)/(x-e)^tanx

Answer 1

Function:\[f(x)=y=\frac{ 1 }{ (x-e)^{\tan x} }=(x-e)^{-tan x}\]I would try logarithmic differentiation (take the ln on both sides, then differentiate, then solve for y'):\[\ln y=\ln(x-e)^{-\tan x}\]\[\ln y = -\tan x \cdot \ln(x-e)\]
Now differentiate both sides (using Chain & Product Rules):\[\ln y \cdot y'=-\sec^2x \cdot \ln(x-e)+-\tan x \cdot \frac{ 1 }{ x-e }\]

Answer 2

Now divide doth sides by lny = -tan x ln(x-e) and try to simplify the formula (can write it as one fraction, but then it's still a bit messy...)