# Section 7.2: Partial Derivatives

## Partial Derivatives

- The
**partial derivative of ***f* with respect to *x*, denoted ∂z/∂x, *f*_{x}, or *f*_{x}(x, y), is defined by:
*∂z/∂x* = limh → 0 *[f(x + h, y) - f(x, y)]/h*

- The
**partial derivative of ***f* with respect to *y*, denoted ∂z/∂y, *f*_{y}, or *f*_{y}(x, y), is defined by:
*∂z/∂y* = limk → 0 *[f(x, y + k) - f(x, y)]/h*

## Second-Order Partial Derivatives

If *z = f(x, y)*, then:

f_{xx} | = f_{xx}(x, y) | = ∂^{2}z/∂x^{2} | = ∂/∂x(∂z/∂x) |

f_{xy} | = f_{xy}(x, y) | = ∂^{2}z/∂y∂x | = ∂/∂y(∂z/∂x) |

f_{yx} | = f_{yx}(x, y) | = ∂^{2}z/∂x∂y | = ∂/∂x(∂z/∂y) |

f_{yy} | = f_{yy}(x, y) | = ∂^{2}z/∂y^{2} | = ∂/∂y(∂z/∂y) |