[tex]y=4e^{\tan x}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm d}{\mathrm dx}[4e^{\tan x}][/tex]
Factor out the constant:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4\dfrac{\mathrm d}{\mathrm dx}[e^{\tan x}][/tex]
Apply the chain rule:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4e^{\tan x}\dfrac{\mathrm d}{\mathrm dx}[\tan x][/tex]
Wrap it up:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4e^{\tan x}\sec^2x[/tex]
