Olá a todos!
Suponha que z = f(x,y) onde x = g(s,t) e y = h(s,t). Admitindo derivadas mistas de segunda ordem contínuas, calcule \(\frac{\partial ^{2}z}{\partial\,t^2 }\).
Estou travando feio numa parte da resolução e gostaria de ajuda para prosseguir
\(\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}\)
\(\frac{\partial ^{2}z}{\partial\,t^2 } = \frac{\partial}{\partial t}\left ( \frac{\partial z}{\partial x} \right )\left ( \frac{\partial x}{\partial t} \right )+\left ( \frac{\partial z}{\partial x} \right )\frac{\partial }{\partial t}\left ( \frac{\partial x}{\partial t} \right )+\frac{\partial }{\partial t}\left ( \frac{\partial z}{\partial y} \right )\left ( \frac{\partial y}{\partial t}\right )+\frac{\partial z}{\partial y}\frac{\partial }{\partial t}\left ( \frac{\partial y}{\partial t} \right )\)
Obrigado a quem puder ajudar.