Introduction to Partial Derivatives
In single-variable calculus, we study how a function f(x) changes as the input variable x changes. The derivative f'(x) or df/dx measures this rate of change. But what happens when a function depends on multiple variables?
Multivariable calculus extends these concepts to functions of several variables, such as f(x,y), f(x,y,z), or even more variables. These functions represent surfaces or hypersurfaces rather than curves.
Key Concept: Partial Derivatives
A partial derivative measures how a multivariable function changes when we vary just one of its variables while holding all others constant.
Notation for Partial Derivatives
For a function f(x,y), we denote the partial derivatives as:
- ∂f/∂x or fx(x,y): the partial derivative with respect to x
- ∂f/∂y or fy(x,y): the partial derivative with respect to y
The symbol ∂ (read as "partial") distinguishes partial derivatives from ordinary derivatives.
Geometric Interpretation
For a function f(x,y) representing a surface in 3D space:
- ∂f/∂x represents the slope of the surface in the x-direction (along a line where y is constant)
- ∂f/∂y represents the slope of the surface in the y-direction (along a line where x is constant)
Computing Partial Derivatives
To find ∂f/∂x, treat y as a constant and differentiate with respect to x using the rules of single-variable calculus. Similarly, to find ∂f/∂y, treat x as a constant and differentiate with respect to y.