The density of a thin circular plate of radius 4 is given by \( \rho(x, y)=4+x y \). The edge of the plate is described by the parametric equations \( x=4 \cos t, y=4 \sin t \), for \( 0 \leq t \leq 2 \pi \). a. Find the rate of change of the density with respect to \( t \) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum? First, find some intermediate derivatives. \( \frac{\partial \rho}{\partial x}=\square \) (Type an expression using \( x \) and \( y \) as the variables.)

To find the partial derivative of the density \( \rho(x, y) = 4 + xy \) with respect to \( x \), we need to differentiate the density function. The expression becomes: \[ \frac{\partial \rho}{\partial x} = y \] Now let’s find the partial derivative of the density with respect to \( y \): \[ \frac{\partial \rho}{\partial y} = x \] With these intermediate derivatives, you can proceed to the next steps for the rate of change of density along the edge of the plate. In part (a), you'll want to find \( \frac{d\rho}{dt} \) using the chain rule, which involves both \( \frac{\partial \rho}{\partial x} \) and \( \frac{\partial \rho}{\partial y} \). For part (b), to find where the density is maximized, evaluate \( \rho \) on the edge using the parametric equations and find critical points in terms of \( t \). Explore the range of density values to identify maximum points.