例題 次の関数の偏導関数$z_{t}$を求めよ.

$\displaystyle z = x^{2} + xy + 2y^{2}, x = \cos{t}, y = \sin{t} $

解答 $z$$x$$y$の関数.$x$$y$$t$の関数より,$z$$x$$y$の変化による影響を受け,$x$$y$$t$の変化の影響を受ける. そこで,

$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$

これより,
$\displaystyle \frac{dz}{dt}$ $\displaystyle =$ $\displaystyle (2x+y)(-\sin{t}) + (x+4y)(\cos{t})$  
  $\displaystyle =$ $\displaystyle (2\cos{t}+\sin{t})(-\sin{t}) + (\cos{t} + 4\sin{t})(\cos{t})$  
  $\displaystyle =$ $\displaystyle -2\cos{t}\sin{t}-\sin^{2}{t}+\cos^{2}{t}+4\sin{t}\cos{t}$  
  $\displaystyle =$ $\displaystyle -\sin^{2}{t}+2\sin{t}\cos{t}+\cos^{2}{t}$