Section 14.5 Chain Rule

Suppose that \(z = f(x, y)\) is a differentiable function of \(x\) and \(y\), where \(x = g(s, t)\) and \(y = h(s, t)\) are differentiable functions of \(s\) and \(t\). Then \[ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \] \[ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \]

EXAMPLE 3 If \(z = e^x \sin y\), where \(x = st^2\) and \(y = s^2t\), find \(\partial z / \partial s\) and \(\partial z / \partial t\).