Given that
#f(x)=x^4\tan x#
Differentiating above function w.r.t. #x# using product rule as follows
#d/dxf(x)=d/dx(x^4\tan x)#
#f'(x)=x^4d/dx\tan x+\tan xd/dxx^4#
#=x^4\sec^2x+\tan x(4x^3)#
#=x^4\sec^2x+4x^3\tan x#
Differentiating above equation w.r.t. #x# using product rule as follows
#d/dxf'(x)=d/dx(x^4\sec^2x+4x^3\tan x)#
#f''(x)=x^4d/dx(sec^2x)+\sec^2xd/dx(x^4)+4x^3d/dx(\tan x)+4\tan xd/dx(x^3)#
#=x^4(2\sec x\sec x\tan x)+\sec^2x(4x^3)+4x^3(\sec^2x)+4\tan x(3x^2)#
#=2x^4\sec^2x\tan x+4x^3\sec^2x+4x^3\sec^2x+12x^2\tan x#
#=2x^4\sec^2x\tan x+8x^3\sec^2x+12x^2\tan x#
