Partial Derivative Rules

Partial derivatives help us understand how functions with multiple variables change when one variable is adjusted. This concept is crucial in Calculus IV, allowing us to analyze complex relationships and behaviors in multivariable functions through various rules and notations.

1. Definition of partial derivatives

   • A partial derivative measures how a function changes as one variable changes while keeping other variables constant.
   • It is used for functions of multiple variables, allowing for the analysis of each variable's effect independently.
   • Denoted as ∂f/∂x, it indicates the derivative of function f with respect to variable x.

2. Partial derivative notation

   • Common notations include ∂f/∂x, ∂f/∂y, and ∂²f/∂x∂y for first and mixed second derivatives.
   • The notation emphasizes the variable of differentiation, distinguishing it from total derivatives.
   • Higher-order partial derivatives can be denoted with multiple partial symbols, e.g., ∂³f/∂x²∂y.

3. Partial derivatives of multivariable functions

   • For a function f(x, y), the partial derivatives with respect to x and y are calculated separately.
   • The result provides insight into the function's behavior in the direction of each variable.
   • Partial derivatives can be used to find critical points and analyze local extrema.

4. Chain rule for partial derivatives

   • The chain rule allows for the differentiation of composite functions involving multiple variables.
   • It states that the derivative of a function with respect to one variable can be expressed in terms of derivatives with respect to other variables.
   • Essential for functions defined implicitly or when variables depend on other variables.

5. Clairaut's theorem (equality of mixed partials)

   • States that if the mixed partial derivatives of a function are continuous, then the order of differentiation does not matter.
   • Formally, ∂²f/∂x∂y = ∂²f/∂y∂x.
   • This theorem simplifies calculations and ensures consistency in multivariable calculus.

6. Implicit differentiation for multivariable functions

   • Used when a function is defined implicitly by an equation involving multiple variables.
   • Allows for finding partial derivatives without explicitly solving for one variable in terms of others.
   • Involves differentiating both sides of the equation with respect to the variable of interest.

7. Gradient vector

   • The gradient vector, denoted as ∇f, consists of all first-order partial derivatives of a function.
   • It points in the direction of the steepest ascent of the function.
   • The magnitude of the gradient indicates the rate of change in that direction.

8. Directional derivatives

   • The directional derivative measures the rate of change of a function in a specified direction.
   • It is calculated using the gradient vector and a unit vector indicating the direction.
   • Provides insight into how the function behaves along arbitrary paths in its domain.

9. Partial derivatives of vector-valued functions

   • For functions that output vectors, partial derivatives are taken with respect to each input variable.
   • Each component of the output vector can have its own partial derivative.
   • Important in fields like physics and engineering where vector fields are analyzed.

10. Higher-order partial derivatives

    • Higher-order partial derivatives involve taking the partial derivative of a partial derivative.
    • They are useful for analyzing the curvature and behavior of multivariable functions.
    • Notation includes second derivatives (∂²f/∂x²) and mixed derivatives (∂²f/∂x∂y).