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4.6 Partial differentiation of composite functions

1.

(a) $\displaystyle{\frac{2x}{x^2 + y^2}(1 - \frac{1}{t^2}) + \frac{2y}{x^2 + y^2}(2t - 1)}$(b) $\displaystyle{z_{x}(2t) + z_{y}e^{t}}$

(c) $\displaystyle{2z_{x} + 8tz_{y}}$(d) $\displaystyle{-6\sin{t}\cos{t}}$

2.

(a) $\displaystyle{z_{r} = \frac{-3s}{r^2 + s^2}, z_{s} = \frac{3r}{r^2 + s^2}}$

(b) $\displaystyle{z_{r} = \frac{-2(r-1)}{(r-1)^2 + s^2} + \frac{2(r+1)}{(r+1)^2 + s^2}, z_{s} = \frac{-2s}{(r-1)^2 + s^2} + \frac{2s}{(r+1)^2 + s^2}}$

(c) $\displaystyle{z_{r} = 1, z_{s} = -2r\sin{s}\cos{s}}$

|bf 3.


$\displaystyle \frac{\partial z}{\partial r}$ $\displaystyle =$ $\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial r}$  
  $\displaystyle =$ $\displaystyle z_{x}\cos{\theta} + z_{y}\sin{\theta}$  
$\displaystyle \frac{\partial z}{\partial \theta}$ $\displaystyle =$ $\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial \theta} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial \theta}$  
  $\displaystyle =$ $\displaystyle z_{x}(-r\sin{\theta}) + z_{y}(r\cos{\theta})$  
  $\displaystyle =$ $\displaystyle r(-z_{x}\sin{\theta} + z_{y}\cos{\theta})$