Differentiation of composite functions

Exercise

1.

Find $${\frac{dz}{dt}}$$．where $f$ is in $C^{1}$．

(a)

$${z = \log{(x^2 + y^2)}, x = t + \frac{1}{t}, y = t(t-1)}$$

(b)

$${z = f(t^2,e^t)}$$

(c)

$${z = f(2t, 4t^2)}$$

(d)

$${z = x^2 - 2y^2, x = \cos{t}, y = \sin{t}}$$

2.

Find $${\frac{\partial z}{\partial r}, \ \frac{\partial z}{\partial s}}$$.

(a)

$${z = \tan^{-1}{\frac{y}{x}}, x = r^3 - 3rs^2, \ y = 3r^2 s - s^3}$$

(b)

$${z = \log{\frac{y}{x}}, x = (r-1)^2 + s^2, y = (r+1)^2 + s^2}$$

3.

For $${z = f(x,y), x = r\cos{\theta}, y = r\sin{\theta}}$$g，show the following．

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