Question

como derivar essa equação y=sen(sen(senx))

Answer 1

Perceba que y é uma função composta. Se, por exemplo, admitirmos g(x)=sen(x), podemos escrever y como:

$\sf \boxed{\sf y~=~g\Big(g\big(g(x)\big)\Big)}~~ou, simplesmente~~\boxed{\sf y~=~g\circ g\circ g}$

Vamos utilizar então a regra da cadeia:

$\sf Para:~~\left\{\begin{array}{l}\sf u~=~sen(sen(x))\\\sf w~=~sen(x)\end{array}\right.\\\\\\\sf \dfrac{dy}{dx}~=~\dfrac{d}{dx}~sen(u)\\\\\\\dfrac{dy}{dx}~=~\dfrac{d}{du}~sen(u)\cdot \dfrac{du}{dx}\\\\\\\dfrac{dy}{dx}~=~\dfrac{d}{du}~sen(u)\cdot \dfrac{d}{dw}~sen(w)\cdot \dfrac{dw}{dx}\\\\\\\dfrac{dy}{dx}~=~\dfrac{d}{du}~sen(u)\cdot \dfrac{d}{dw}~sen(w)\cdot \dfrac{d}{dx}~sen(x)$

$\sf \dfrac{dy}{dx}~=~cos(u)\cdot cos(w)\cdot cos(x)\\\\\\\boxed{\sf \dfrac{dy}{dx}~=~cos\Big(sen\big(sen(x)\big)\Big)\cdot cos\big(sen(x)\big)\cdot cos(x)}$

$\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$