The chain rule is:
#(d[f(g(x))])/dx = ((df(g))/(dg))((dg(x))/dx)#
Let #g(x) = sin(x^3)#, then #f(g) = cos^-1(g)# and #(df(g))/(dg) = -1/sqrt(1 - g^2#
For #(dg(x))/dx#, use the chain rule, again:
#(d[g(h(x))])/dx = ((dg(h))/(dh))((dh(x))/dx)#
Let #h(x) = x^3#, then #g(h) = sin(h)#, #(dg(h))/(dh) = cos(h)#, and #(dh(x))/dx = 3x^2#
Substitute back into the second chain rule:
#(dg(x))/dx = 3x^2cos(h)#
Reverse the substitution for h:
#(dg(x))/dx = 3x^2cos(x^3)#
Substitute back into the first chain rule:
#(d[f(g(x))])/dx = -3x^2cos(x^3)/sqrt(1 - g^2)#
Reverse the substitution for g:
#(d[cos^-1(sin(x^3))])/dx = -3x^2cos(x^3)/sqrt(1 - sin^2(x^3))#
Please observe that #sqrt(1 - sin^2(x^3)) = cos(x^3)# which causes the fraction to become 1:
#(d[cos^-1(sin(x^3))])/dx = -3x^2#
