Question

Use la regla de la cadena para calcular ∂z/∂s y ∂z/∂t donde z = xy +y^2-7xy^3,
x = s sin(t) y y = t sin(s).

Answer 1

$\dfrac{\partial z}{\partial s}=\dfrac{\partial z}{\partial x}\cdot\dfrac{\partial x}{\partial s}+\dfrac{\partial z}{\partial y}\cdot\dfrac{\partial y}{\partial s}\\ \\ \\ \dfrac{\partial z}{\partial s}=(y-7y^3)\cdot(\sin t)+(x+2y-21xy^2)\cdot(t\cos s)\\ \\ \\ \texttt{Lo dem\'as es reemplazar y \'algebra elemental}$


$\dfrac{\partial z}{\partial t}=\dfrac{\partial z}{\partial x}\cdot\dfrac{\partial x}{\partial t}+\dfrac{\partial z}{\partial y}\cdot\dfrac{\partial y}{\partial t}\\ \\ \\ \dfrac{\partial z}{\partial t}=(y-7y^3)\cdot(s\cos t)+(x+2y-21xy^2)\cdot (\sin s)$