Use the quotient rule to differentiate: ln(3x)/(e^4x) with respect to x.

Quotient rule: d(u/v)/dx = [(du/dx)v-u(dv/dx)]/v^2
u = ln(3x)
v = e^4x
Find du/dx using chain rule:
u = ln(z) ==> du/dz = 1/z
z = 3x ==> dz/dx = 3
(du/dz)(dz/dx) = 3/z = 3/3x = 1/x
du/dx = 1/x
Find dv/dx
v = e^4x ==> dv/dx = 4e^4x
Plugging into the quotient rule equation:
d(u/v)/dx = [(du/dx)v-u(dv/dx)]/v^2
[(1/x)e^4x - ln(3x)(4e^4x)]/(e^4x)^2
Simplify:
[e^4x ((1/x)- 4ln(3x))]/(e^4x)^2
[(1/x)- 4ln(3x)]/(e^4x)
Final answer: d/dx = [(1/x)- 4ln(3x)]/(e^4x)