Essential Concepts
- A partial derivative is a derivative involving a function of more than one independent variable.
- To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
- Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.
Key Equations
- Partial derivative of [latex]f[/latex] with respect to [latex]x[/latex]
[latex]\frac{\partial f}{\partial x}=\underset{h \to{0}}{\lim} \frac{f(x+h,y)-f(x,y)}{h}[/latex] - Partial derivative of [latex]f[/latex] with respect to [latex]y[/latex]
[latex]\frac{\partial f}{\partial y}=\underset{k \to{0}}{\lim} \frac{f(x,y+k)-f(x,y)}{k}[/latex]
Glossary
- higher-order partial derivatives
- second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives
- mixed partial derivatives
- second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables
- partial derivative
- a derivative of a function of more than one independent variable in which all the variables but one are held constant
- partial differential equation
- an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives
