Problems
1. Suppose $f$ is a function of $x$, $y$, and $z$ as follows
$f(x, y, z) = axy + by^2 + cxz$
where $a$, $b$, and $c$ are constants. Furthermore, $x$ and $y$ are functions of $z$ as follows
$x(z) = bz$
$y(z) = cz$
(a) Compute the three partial derivatives:
$\frac{\partial f}{\partial x} \bigg|_{y,z}$
$\frac{\partial f}{\partial y} \bigg|_{x,z}$
$\frac{\partial f}{\partial z} \bigg|_{x,y}$
(b) Compute the total derivative $\frac{df}{dz}$ using the chain rule formula:
$\frac{df}{dz} = \frac{\partial f}{\partial x} \frac{dx}{dz} + \frac{\partial f}{\partial y} \frac{dy}{dz} + \frac{\partial f}{\partial z}$
(c) Using the equations for $x$ and $y$ in terms of $z$, express $f$ in terms of $z$ only; i.e., write the
function as $f(z)$.
(d) Compute the total derivative $\frac{df}{dz}$ directly using the result of part (c) and verify that the
result matches that found in part (b).
