Question

Calcule a derivada de cada uma das funções abaixo:

Answer 1

item a)

$\displaystyle \frac{\text {df}}{\text {dx}} \to\text{f(x)}=\text{arctan}(\text x^2-4\text x) \\\\ \text{Sendo u = }\text x^2-4\text x : \\\\ \text{f'(x)}=\frac{\text u'}{1+\text u^2} \\\\\\ \text{f'(x)} = \frac{(\text x^2-4\text x)'}{1+(\text x^2-4\text x)^2} \\\\\\ \huge\boxed{\text{f'(x)}=\frac{2\text x-4 }{1+(\text x^2-4\text x)^2}\ }\checkmark$

item b)

$\text{f(x)}=\text{x}^{\text{cos x} }$

Vamos usar um macete para esse tipo de exponencial que é aplicar logaritmo natural em ambos os lados e derivar :

$\displaystyle \text{ln f(x)} = \text{ln x}^{\text{cos x}} \\\\ \text{ln f(x)}=(\text{cos x}).\text{ln(x)} \\\\ \underline{\text{Derivando}}: \\\\ \frac{\text{f'(x)}}{\text{f(x)}}= -(\text{sen x}).\text{ln(x)}+(\text{cos x})\frac{1}{\text x} \\\\\\ \boxed{\text{f'(x)} =\text x^{\displaystyle \text{cos x}} [\ -\text{sen (x)}.\text{ln(x)}.+\frac{\text{cos (x)}}{\text x}\ ] . }\checkmark$