Lecture Notes on Partial Derivatives

Introduction to Multivariable Functions

• Example of a multivariable function:
  • f(x, y) = x²y + sin(y)
• Output is a scalar value.

Understanding Derivatives

Ordinary Derivatives

• Notation: df/dx, where f(x) = x².
• Graphical Interpretation:
  • Input (x-axis) and Output (y-axis).
  • Example:
    • At x = 2, a small change dx results in a change df.
    • Relation: slope = rise/run.
• Optionally visualized on number lines instead of graphs.

Multivariable Derivatives

• Partial Derivative Notation:
  • ∂f/∂x
  • Treats one variable as constant while differentiating the other.
• Visualized on the xy-plane:
  • Input space represented as the xy-plane.
  • Evaluating at a point (1, 2).
  • Change in x (dx) influences the output, yielding df.
  • Change in y (dy) similarly affects output.

Notation for Partial Derivatives

• New notation emphasizes the multivariable nature:
  • Example: ∂f/∂y.
• "Partial" indicates that the derivative accounts for change in one variable at a time.

Example Calculations

Evaluating ∂f/∂x at (1, 2)

• Keep y constant at 2:
  • f(x, y) = x²(2) + sin(2).
• Derivative:
  • ∂f/∂x = 4x
  • Result at x=1:
    • Answer = 4.

Evaluating ∂f/∂y at (1, 2)

• Keep x constant at 1:
  • f(1, y) = (1²)y + sin(y).
• Derivative:
  • ∂f/∂y = 1 + cos(y).
  • Result at y=2:
    • Answer = 1 + cos(2).

General Formula Derivation

∂f/∂x as a Function of x and y

• Treat y as constant:
  • ∂f/∂x = 2xy + 0 = 2xy (when evaluated).

∂f/∂y as a Function of x and y

• Treat x as constant:
  • ∂f/∂y = x² + cos(y).

Conclusion

• Partial derivatives help in understanding how a function behaves with respect to each variable while keeping others constant.
• Important for visualizing and studying functions with multiple inputs.
• Upcoming discussions will include graphical interpretations and higher-dimensional functions.