Question

sejam f(x)=sen^2 x e g(x)=cos^2 x. O valor de f'(x)+g'(x)é?

Answer 1

Explicação passo-a-passo:

Temos que $f(x)=sen^{2}(x)$ e $g(x)=\cos^{2}(x)$.

Para achar a derivada dessas funções, utilizaremos a Regra da Cadeia:

$y=u(v(x)) \implies y'=u'(v(x)).v'(x)$

Derivando f(x):

$f(x)=sen^{2}(x)\implies u(x)=x^{2} \ e\ v(x)=sen(x) \\\\u'(x)=2.x\ e\ v'(x)=cos(x)$

Aplicando a Regra da Cadeia:

$f'(x)=u'(v(x)).v'(x)\\\\f'(x)=u'(sen(x)).v'(x)\\\\\boxed{f'(x)=2.sen(x).cos(x) \iff f'(x)=sen(2x)}$

Derivando g(x):

$g(x)=\cos^{2}(x) \implies u(x)=x^{2} \ e\ v(x)=\cos(x)\\\\u'(x)=2.x\ e\ v'(x)=-sen(x)$

Aplicando a Regra da Cadeia:

$g'(x)=u'(v(x)).v'(x)\\\\g'(x)=u'(\cos(x)).v'(x)\\\\g'(x)=2.\cos(x).(-sen(x))\\\\\boxed{g'(x)=-2.sen(x).cos(x)\iff g'(x)=-sen(2x)}$

Calculando f'(x)+g'(x):

$f'(x)+g'(x)=sen(2x)+(-sen(2x))\\\\\boxed{\boxed{f'(x)+g'(x)=0}}$