# Second Order Partial Derivatives in Calculus

Examples with detailed solutions on how to calculate second order partial derivatives are presented.

## Definitions and Notations of Second Order Partial Derivatives

For a two variable function $$f(x , y)$$, we can define 4 second order partial derivatives along with their notations.

## Examples with Detailed Solutions on Second Order Partial Derivatives

### Example 1

Find $$f_{xx}$$, $$f_{yy}$$ given that $$f(x , y) = \sin(x y)$$.

#### $$f_{xx}$$ may be calculated as follows

$f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin (x y)\right) = \frac{\partial}{\partial x}(y \cos (x y)) = - y^2 \sin (x y)$

#### $$f_{yy}$$ can be calculated as follows

$f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin (x y)\right) = \frac{\partial}{\partial y}(x \cos (x y)) = - x^2 \sin (x y)$

### Example 2

Find $$f_{xx}$$, $$f_{yy}$$, $$f_{xy}$$, $$f_{yx}$$ given that $$f(x , y) = x^3 + 2 x y$$.

#### $$f_{xx}$$ is calculated as follows

$f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2 x y)\right) = \frac{\partial}{\partial x}(3 x^2 + 2 y) = 6x$

#### $$f_{yy}$$ is calculated as follows

$f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2 x y)\right) = \frac{\partial}{\partial y}(2x) = 0$

#### $$f_{xy}$$ is calculated as follows

$f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2 x y)\right) = \frac{\partial}{\partial y}(3 x^2 + 2 y) = 2$

#### $$f_{yx}$$ is calculated as follows

$f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2 x y)\right) = \frac{\partial}{\partial x}(2x) = 2$

### Example 3

Find $$f_{xx}$$, $$f_{yy}$$, $$f_{xy}$$, $$f_{yx}$$ given that $$f(x , y) = x^3y^4 + x^2 y$$.

#### $$f_{xx}$$ is calculated as follows

$f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2 y)\right) = \frac{\partial}{\partial x}(3 x^2y^4 + 2 x y) = 6xy^4 + 2 y$

#### $$f_{yy}$$ is calculated as follows

$f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2 y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2$

#### $$f_{xy}$$ is calculated as follows

$f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2 y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2 x y) = 12x^2y^3 + 2x$

#### $$f_{yx}$$ is calculated as follows

$f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2 y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x$

## More References and Links to Partial Derivatives and Multivariable Functions

Multivariable Functions