Question

Suponha que $g'(3) = -5$, $f(2) =3$, $f'(2) = 4$ e $g(3) =6$. Assinale o item que corresponde a $h'(2)$ em que $h(x) = g(f(x))$:
$a) -15 \: \: \: b)-20 \: \: \: c)18 \: \: \: d)24 \: \: \: e)-30$​

Answer 1

Regra da cadeia :

$\text{[ f(u) ]'} = \text{f '(u)}.\text{u'} \\\\ \underline{\text{onde}} : \text{u {\'e} uma fun{\c c}{\~a}o}}$

Temos :

$\text{h(x)}=\text{g(f(x)) } \\\\ \underline{\text{Derivando}}: \\\\ \text{h'(x)}=\text{g'(f(x)).f '(x)} \\\\ \underline{\text{fazendo x = 2}}: \\\\\ \text{h'(2)}=\text{g'(f(2)).f'(2)} \\\\ \underline{\text{temos}}: \\\\ \text{f(2) = 3 \ ; \ g'(3) }=-5 \ \ ; \ \text {f '(2) =4 } \\\\ \underline{\text{substituindo}}: \\\\\ \text{h'(2)}=\text {g'(3).4}\\\\ \text{h'(2)}=-5.4 \\\\ \huge\boxed{\ \text{h'(2)} = -20\ }\checkmark$

letra b