Section 14.5 Chain Rule
Example
If \(z = f(x, y)\) has continuous second-order partial derivatives and \(x = r^2 + s^2\) and \(y = 2rs\), find (a) \(\partial z / \partial r\) and (b) \(\partial^2 z / \partial r^2\).
SOLUTION (a) The Chain Rule gives \[ \frac{\partial z}{\partial r} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial r} = \frac{\partial z}{\partial x}(2r) + \frac{\partial z}{\partial y}(2s) \] (b) Applying the Product Rule to the expression in part (a), we get \[ \frac{\partial^2 z}{\partial r^2} = \frac{\partial}{\partial r} \left( 2r \frac{\partial z}{\partial x} + 2s \frac{\partial z}{\partial y} \right) \] \[ = 2 \frac{\partial z}{\partial x} + 2r \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial x} \right) + 2s \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial y} \right) \] But, using the Chain Rule again (see Figure 5), we have \[ \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial x} \right) = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial x} \right) \frac{\partial x}{\partial r} + \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) \frac{\partial y}{\partial r} = \frac{\partial^2 z}{\partial x^2}(2r) + \frac{\partial^2 z}{\partial y \partial x}(2s) \] \[ \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial y} \right) = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) \frac{\partial x}{\partial r} + \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial y} \right) \frac{\partial y}{\partial r} = \frac{\partial^2 z}{\partial x \partial y}(2r) + \frac{\partial^2 z}{\partial y^2}(2s) \] Putting these expressions into Equation 5 and using the equality of the mixed second-order derivatives, we obtain \[ \frac{\partial^2 z}{\partial r^2} = 2 \frac{\partial z}{\partial x} + 2r \left( 2r \frac{\partial^2 z}{\partial x^2} + 2s \frac{\partial^2 z}{\partial y \partial x} \right) + 2s \left( 2r \frac{\partial^2 z}{\partial x \partial y} + 2s \frac{\partial^2 z}{\partial y^2} \right) \] \[ = 2 \frac{\partial z}{\partial x} + 4r^2 \frac{\partial^2 z}{\partial x^2} + 8rs \frac{\partial^2 z}{\partial x \partial y} + 4s^2 \frac{\partial^2 z}{\partial y^2} \]