Question 1
What is the chain rule for a function $z = f(x, y)$ where $x = g(t)$ and $y = h(t)$?
Question 2
Given $z = x^2 + 3xy + y^2$ where $x = t^2$ and $y = \sin t$, what is $\frac{dz}{dt}$?
Question 3
If $w = f(x, y, z)$ and $x = u + v$, $y = u - v$, $z = u v$, what is $\frac{\partial w}{\partial u}$ using the chain rule?
Question 4
Consider the implicit equation $x^2 + y^2 + z^2 = 1$. Find $\frac{\partial z}{\partial x}$ at the point $(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2})$ assuming $z > 0$.
Question 5
If $w = e^{xy}$ and $x = t^2 + 1$, $y = \sin t$, what is $\frac{dw}{dt}$?
