If #f(x) =-e^(-x) # and #g(x) = tan^2x^2 #, what is #f'(g(x)) #?
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#f' (g(x))=4x*e^(-tan^2 x^2)*(tan^3 x^2+tan x^2)#
given #f(x)=-e^(-x)# and #g(x)=tan^2 x^2#
#f(g(x))=-e^(-g(x))#
#f(g(x))=-e^(-tan^2 x^2)#
#f' (g(x))=(-1)*d/dx(e^(-tan^2 x^2))#
#f' (g(x))=(-1)*(e^(-tan^2 x^2))*d/dx(-tan^2 x^2)#
#f' (g(x))=(-1)*(e^(-tan^2 x^2))*-d/dx(tan^2 x^2)#
#f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))d/dx(tan x^2)#
#f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))(sec^2 x^2)d/dx(x^2)#
#f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))(sec^2 x^2)(2x)#
Simplify at this point
#f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan x^2)(sec^2 x^2)#
#f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan x^2)(tan^2 x^2+1)#
#f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan^3 x^2+tan x^2)#
God bless....I hope the explanation is useful.
