Question

Se f(1) = 0, f'(1) = 2g(2)g(3) = 1 e g'(3) = 3/2, calcule (fog)'(3). Obrigada a quem ajudar.

Answer 1

Datos con lo que he resuelto
Sea f(1) = 0, f ' (1) = 2, g' (3) = 3/2

Aplicar de la cadena
(fog)'(x) = {f[g(x)]}'

sea y = g(x)
$\dfrac{d}{dx}(f\circ g)(x)= \dfrac{d}{dx} g(x)\cdot \dfrac{d}{dy}f(y)$

Evaluando en x = 3
        $\dfrac{d}{dx}(f\circ g)(3)= \dfrac{d}{dx} g(3)\cdot \dfrac{d}{dy}f(g(3))\\ \\ \dfrac{d}{dx}(f\circ g)(3)= g'(3)\cdot \dfrac{d}{dy}f(1)\\ \\ \dfrac{d}{dx}(f\circ g)(3)=\dfrac{3}{2}\cdot f'(1)\\ \\ \dfrac{d}{dx}(f\circ g)(3)=\dfrac{3}{2}\cdot 2\\ \\ \\ \boxed{\dfrac{d}{dx}(f\circ g)(3)=3}$