Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.3 Derivatives Of Inverse Functions; Derivatives And Integrals Involving Exponential Functions - Exercises Set 6.3 - Page 432: 21

$$y' = {e^{x\tan x}}\left( {x{{\sec }^2}x + \tan x} \right)$$

$$\eqalign{ & y = {e^{x\tan x}} \cr & {\text{differentiate}} \cr & y' = \left( {{e^{x\tan x}}} \right)' \cr & {\text{use the chain rule}} \cr & y' = {e^{x\tan x}}\left( {x\tan x} \right)' \cr & {\text{product rule}} \cr & y' = {e^{x\tan x}}\left( {x\left( {\tan x} \right)' + \tan x\left( x \right)'} \right) \cr & y' = {e^{x\tan x}}\left( {x{{\sec }^2}x + \tan x\left( 1 \right)} \right) \cr & {\text{Simplify}} \cr & y' = {e^{x\tan x}}\left( {x{{\sec }^2}x + \tan x} \right) \cr} $$