4.6 Partial differentiation of composite functions

1.

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

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

2.

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

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

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

|bf 3.

$$\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial r}$$

$$= z_{x}\cos{\theta} + z_{y}\sin{\theta}$$

$$\frac{\partial z}{\partial \theta} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial \theta} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial \theta}$$

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

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