1. Understanding Partial Derivatives
Partial derivatives are used when dealing with functions of more than one variable. They measure how the function changes as only one variable is varied, keeping the others constant.
If z = f(x, y), then the partial derivatives of f with respect to x and y are:
- ∂f/∂x = rate of change of f with respect to x (y held constant)
- ∂f/∂y = rate of change of f with respect to y (x held constant)
2. Notation and Example
Different notations for partial derivatives include:
- ∂f/∂x or fx
- ∂f/∂y or fy
Example: If f(x, y) = x²y + 3xy², then:
- ∂f/∂x = 2xy + 3y²
- ∂f/∂y = x² + 6xy
3. Higher-Order Partial Derivatives
Just like ordinary derivatives, partial derivatives can also be taken multiple times.
For f(x, y), the second-order partial derivatives include:
- ∂²f/∂x² (second partial with respect to x)
- ∂²f/∂y² (second partial with respect to y)
- ∂²f/∂x∂y and ∂²f/∂y∂x (mixed partials, often equal if f is continuous)