EXERCISES 14.4
Chain Rule: One Independent Variable
In Exercises 1-6, (a) express $\frac{dw}{dt}$ as a function of $t$, both by using
the Chain Rule and by expressing $w$ in terms of $t$ and differentiating
directly with respect to $t$. Then (b) evaluate $\frac{dw}{dt}$ at the given value
of $t$.
1. $w = x^2 + y^2$, $x = \cos t$, $y = \sin t$; $t = \pi$
2. $w = x^2 + y^2$, $x = \cos t + \sin t$, $y = \cos t - \sin t$; $t = 0$
3. $w = \frac{x}{z} + \frac{y}{z}$, $x = \cos^2 t$, $y = \sin^2 t$, $z = 1/t$; $t = 3$
4. $w = \ln (x^2 + y^2 + z^2)$, $x = \cos t$, $y = \sin t$, $z = 4\sqrt{t}$;
$t = 3$
