Higher Partial Derivatives

The first partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ are again functions of $x$ and $y$ and can be differentiated again with respect to each variable. These derivatives are called higher derivatives.

A function with $n$ variables has

• $n$ first partial derivatives,
• $n^2$ second partial derivatives,
• $n^3$ third partial derivatives, etc.

Example 2: again: $u(x,y)=5x^2+3xy+y^4$.
We already know:

$$\frac{\partial u}{\partial x}=10x+3y \hspace{1cm}\mbox{and}\hspace{1cm} \frac{\partial u}{\partial y}=3x+4y^3.$$

The second partial derivative of $u$ with respect to $x$ is:

$$\frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial x}(10x+3y)=10,$$ (7)

and, with respect to $y$, is:

$$\frac{\partial^2 u}{\partial y^2}=\frac{\partial u}{\partial y}(3x+4y^3)=12y^2.$$ (8)