[tex]f:\mathbb{R}\to\mathbb{R},\,\, f(x) = x^3+x+2\\ \\ x^3+x+2 = f(x)\Big|f^{-1}()[/tex]
[tex]\Rightarrow f^{-1}(x^3+x+2) = f^{-1}\big(f(x)\big)[/tex]
[tex]\Rightarrow f^{-1}(x^3+x+2) = x\Big|'[/tex]
[tex]\Rightarrow (x^3+x+2)'(f^{-1})'(x^3+x+2) = x'[/tex]
[tex]\Rightarrow (3x^2+1)(f^{-1})'(x^3+x+2) = 1[/tex]
[tex]\Rightarrow (f^{-1})'(x^3+x+2) = \dfrac{1}{3x^2+1}[/tex]
[tex]\text{Fac }x=1:[/tex]
[tex]\Rightarrow (f^{-1})'(1+1+2)=\dfrac{1}{3+1}[/tex]
[tex]\Rightarrow \boxed{(f^{-1})'(4) = \dfrac{1}{4}}[/tex]
