Answer
$\frac{dz}{dt} = -18$
Work Step by Step
$x^2+y^2+z^2 = 9$
We can differentiate both sides of the equation with respect to $t$:
$2x\frac{dx}{dt}+2y\frac{dy}{dt}+2z\frac{dz}{dt} = 0$
$2z\frac{dz}{dt} = -2x\frac{dx}{dt}-2y\frac{dy}{dt}$
$z\frac{dz}{dt} = -x\frac{dx}{dt}-y\frac{dy}{dt}$
$\frac{dz}{dt} = \frac{-x\frac{dx}{dt}-y\frac{dy}{dt}}{z}$
$\frac{dz}{dt} = \frac{-(2)(5)-(2)(4)}{(1)}$
$\frac{dz}{dt} = -18$