According to the chain rule, #(f(g(x)))'=f'(g(x))*g'(x)#.
We have #f(u)=lnu# where #u=g(x)=sqrt(-e^(4x)-2)#
We need to find #d/(du)lnu*d/(dx)sqrt(-e^(4x)-2)#
Since #d/dxlnx=1/x#, we write:
#1/u(d/(dx)sqrt(-e^(4x)-2))#
Concentrating on the bracket, let's define two other functions:
#h(v)=sqrt(v)# where #v=j(x)=-e^(4x)-2#
Since #d/dxsqrt(x)=1/(2sqrt(x))#, we write:
#1/(2sqrt(v))(d/dx(-e^(4x)-2))#
Concentrating on the bracket, we define two more functions:
#k(z)=-e^z-2#, where #z=l(x)=4x#
We have #d/(dz)(-e^z-2)*d/dx(4x)#
#-(d/(dz)e^z+d/(dz)2)*4d/dxx#
#-(e^z)*4#
#-4e^z#
Overall, we have:
#1/u(1/(2sqrt(v))(-4e^z))#
Well, that's not right, right? We just have a bunch of nonsense variables. But remember that:
#u=sqrt(-e^(4x)-2)#
#v=-e^(4x)-2#
#z=4x#
We can input our stuff in:
#1/sqrt(-e^(4x)-2)(1/(2sqrt(-e^(4x)-2))(-4e^(4x)))#
And simplify:
#1/sqrt(-e^(4x)-2)((-4e^(4x))/(2sqrt(-e^(4x)-2))#
#1/sqrt(-e^(4x)-2)(-(2e^(4x))/sqrt(-e^(4x)-2))#
#-(2e^(4x))/(-e^(4x)-2)#
#-(2e^(4x))/(-1(e^(4x)+2))#
#(2e^(4x))/(e^(4x)+2)#
The answer.
