Question

Sendo: w=x+2y+z² , x= sen(t) , y= cos(t) e z= t-1 , dw/dt , é :

Answer 1

Aqui devemos usar a Regra da Cadeia para funções de várias variáveis:

$w(x,\,y,\,z)=x+2y+z^2$


$\dfrac{dw}{dt}(t)=\dfrac{\partial w}{\partial x}\cdot \dfrac{dx}{dt}+\dfrac{\partial w}{\partial y}\cdot \dfrac{dy}{dt}+\dfrac{\partial w}{\partial z}\cdot \dfrac{dz}{dt}\\\\\\ \dfrac{dw}{dt}(t)=1\cdot \cos t+2\cdot (-\mathrm{sen\,}t)+2z(t)\cdot 1\\\\\\ \dfrac{dw}{dt}(t)=\cos t-2\,\mathrm{sen\,}t+2\cdot (t-1)\\\\\\ \boxed{\begin{array}{c} \dfrac{dw}{dt}(t)=\cos t-2\,\mathrm{sen\,}t+2\,(t-1) \end{array}}$