Definitions and Notations of Second Order Partial Derivatives
Examples with detailed solutions on how to calculate second order partial derivatives are presented.
For a function of two variables \( f(x, y) \), we can define four second-order partial derivatives along with their standard notations:
\[ \begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial x} (f_x) = (f_x)_x = f_{xx} \\\\ \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial y} (f_y) = (f_y)_y = f_{yy} \\\\ \dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial y} (f_x) = (f_x)_y = f_{xy} \\\\ \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial x} (f_y) = (f_y)_x = f_{yx} \end{aligned} \]Examples with Detailed Solutions on Second Order Partial Derivatives
Click on each example to view its detailed step-by-step solution.
Example 1
Find \( f_{xx} \) and \( f_{yy} \) given that \( f(x, y) = \sin(xy) \).
Show Solution to Example 1
Calculating \( f_{xx} \):
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin(xy)\right) = \frac{\partial}{\partial x}(y \cos(xy)) = -y^2 \sin(xy) \]Calculating \( f_{yy} \):
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin(xy)\right) = \frac{\partial}{\partial y}(x \cos(xy)) = -x^2 \sin(xy) \]Example 2
Find \( f_{xx} \), \( f_{yy} \), \( f_{xy} \), and \( f_{yx} \) given that \( f(x, y) = x^3 + 2xy \).
Show Solution to Example 2
Calculating \( f_{xx} \):
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(3x^2 + 2y) = 6x \]Calculating \( f_{yy} \):
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(2x) = 0 \]Calculating \( f_{xy} \):
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(3x^2 + 2y) = 2 \]Calculating \( f_{yx} \):
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(2x) = 2 \]Example 3
Find \( f_{xx} \), \( f_{yy} \), \( f_{xy} \), and \( f_{yx} \) given that \( f(x, y) = x^3y^4 + x^2y \).
Show Solution to Example 3
Calculating \( f_{xx} \):
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(3x^2y^4 + 2xy) = 6xy^4 + 2y \]Calculating \( f_{yy} \):
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2 \]Calculating \( f_{xy} \):
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2xy) = 12x^2y^3 + 2x \]Calculating \( f_{yx} \):
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x \]