Question

DERIVADAS
Se F(x)=1+sen x / 1-sen x , calcule F''(π):

Answer 1

Queremos a segunda derivada aplicada no pi, então, pela regra do quociente, temos que:

$\boxed{(\frac{u}{v})'=\frac{u'v-uv'}{v^2}}$

$f(x) = \frac{1+senx}{1-senx}\\\\ f'(x)=\frac{cosx(1-senx)-(-cosx)(1+senx)}{(1-senx)^2}\\\\ f'(x)=\frac{(cosx-senxcosx+cosx+senxcosx)}{(1-senx)^2}\\\\ \boxed{f'(x)=\frac{2cosx}{(1-senx)^2}}$

Segunda derivada:

$f'(x)=\frac{2cosx}{(1-senx)^2}\\\\ f''(x)=\frac{-2senx(1-senx)^2-(2cosx)(2(1-senx)(-cosx)}{(1-senx)^4}\\\\ \boxed{f''(x)=\frac{-2senx(1-senx)^2-(2cosx)(2(1-senx)(-cosx)}{(1-senx)^4}}$

Substituindo pi

$f''(x)=\frac{-2senx(1-senx)^2-(2cosx)(2(1-senx)(-cosx)}{(1-senx)^4}\\\\ f''(\pi)=\frac{-2sen\pi(1-sen\pi)^2-(2cos\pi)(2(1-sen\pi)(-cos\pi)}{(1-sen\pi)^4}\\\\ f''(\pi)=\frac{-2(0)(1-(0))^2-(2(-1))(2(1-(0))(-(-1))}{(1-(0))^4}\\\\ f''(\pi)=\frac{0+4}{(1)}\\\\ \boxed{f''(\pi)=4}$