Question

Consideram functia f:R->R, f(x)=x^3+x+2. Valoarea derivatei (f^-1)'(4) este egala cu:

Answer 1

$f:\mathbb{R}\to\mathbb{R},\,\, f(x) = x^3+x+2\\ \\ x^3+x+2 = f(x)\Big|f^{-1}()$

$\Rightarrow f^{-1}(x^3+x+2) = f^{-1}\big(f(x)\big)$

$\Rightarrow f^{-1}(x^3+x+2) = x\Big|'$

$\Rightarrow (x^3+x+2)'(f^{-1})'(x^3+x+2) = x'$

$\Rightarrow (3x^2+1)(f^{-1})'(x^3+x+2) = 1$

$\Rightarrow (f^{-1})'(x^3+x+2) = \dfrac{1}{3x^2+1}$

$\text{Fac }x=1:$

$\Rightarrow (f^{-1})'(1+1+2)=\dfrac{1}{3+1}$

$\Rightarrow \boxed{(f^{-1})'(4) = \dfrac{1}{4}}$