Partial Derivative Notation

Partial derivatives are a way to calculate the rate of change of a multivariable function with respect to one of its variables, while holding the other variables constant. This is denoted using partial derivative notation.

Single Variable Derivatives vs. Partial Derivatives

In single-variable calculus, derivatives are denoted by dy/dx, where y is the dependent variable and x is the independent variable. In the case of functions of multiple variables, we use partial derivative notation to indicate which variable we are differentiating with respect to.

Partial Derivative Notation

The partial derivative of a function f with respect to the variable x is denoted by ∂f/∂x or ∂ⁿf/∂xⁿ for higher-order derivatives. The symbol ∂ is used to indicate that we are taking the partial derivative.

Example

For a function f(x, y) = x²y + 3x - y, the partial derivative with respect to x would be denoted as ∂f/∂x = 2xy + 3. This indicates how f changes as x changes, keeping y constant.

Overall, partial derivative notation is essential in multivariable calculus for understanding how functions change with respect to specific variables while holding others constant.