Applying The Chain Rule to Functions of Several Variables Examples 2

We saw some examples of applying the various chain rules for functions of several variables on the Applying The Chain Rule to Functions of Several Variables page. We will now look at some more examples. Once again, here are the generic chain rules for reference:

If $z = f(x, y)$ has continuous first partial derivatives and $x = x(t)$ and $y = y(t)$ are differentiable then:

(1)

\begin{align} \quad \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} \end{align}

If $z = f(x, y)$ is a two variable real-valued function with continuous first partial derivatives, and $x = x(s, t)$ and $y = y(s, t)$ are functions of $s$ and $t$ then:

(2)

\begin{align} \quad \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} \end{align}

(3)

\begin{align} \quad \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} \end{align}

We will now look at some more examples of applying the chain rules.

Example 1

Determine $\frac{\partial^3}{\partial y^2} f(2x + 3y, xy)$ in terms of partial derivatives of $f$.

Let $u = 2x + 3y$ and $v = xy$. Then $f(2x + 3y, xy) = f(u, v)$. We note that $f$ is dependent on $u$ and $v$ and both $u$ and $v$ are dependent on $x$ and $y$. We first compute $\frac{\partial f}{\partial y}$:

(4)

\begin{align} \quad \frac{\partial f}{\partial y} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial y} \\ \quad \frac{\partial f}{\partial y} = 3f_1(u, v) + x f_2(u, v) \end{align}

We now differentiate the result again with respect to $y$:

(5)

\begin{align} \quad \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} \left [ 3f_1(u, v) + x f_2(u, v) \right ] \\ \quad \frac{\partial^2 f}{\partial y^2} = 3 \left ( \frac{\partial f_1}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial f_1}{\partial v} \frac{\partial v}{\partial y} \right ) + (0) f_2(u, v) + x \left ( \frac{\partial f_2}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial f_2}{\partial v} \frac{\partial v}{\partial y} \right ) \\ \quad \frac{\partial^2 f}{\partial y^2} = 3 \left ( 3f_{11}(u, v) + xf_{12} (u, v) \right ) + x \left ( 3f_{21} (u, v) + xf_{22}(u, v) \right ) \end{align}