Section 14.5 Chain Rule

SOLUTION With the help of the tree diagram in Figure 4, we have \[ \frac{\partial u}{\partial s} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial s} \] \[ = (4x^3y)(re^t) + (x^4 + 2yz^3)(2rse^{-t}) + (3y^2z^2)(r^2 \sin t) \] When \(r = 2, s = 1\), and \(t = 0\), we have \(x = 2, y = 2\), and \(z = 0\), so \[ \frac{\partial u}{\partial s} = (64)(2) + (16)(4) + (0)(0) = 192 \]