Partial Derivatives Lecture Notes

Introduction to Partial Derivatives

• Partial Derivative: Measures how a function changes as one variable changes while others are held constant.
• Notation: F subx or F ofx represents partial derivative with respect to X.

Finding Partial Derivatives

• With Respect to X:

  • Treat Y as a constant.
  • Example: For function 7x^2 - x^3y^4 + 5x^4y^3, partial derivative w.r.t X is 14x - 3x^2y^4 + 20x^3y^3.

• With Respect to Y:

  • Treat X as a constant.
  • Example: Partial derivative w.r.t Y for same function is -4xy^3 + 15x^4y^2.

Power Rule Review

• Power Rule: Derivative of x^n is nx^(n-1).
  • Examples: Derivative of x^4 is 4x^3, x^3 is 3x^2, x^2 is 2x.

Exponential Functions

• Derivative Rule: Derivative of e^u is e^u * u'.
  • Example: Derivative of e^(x^3) is e^(x^3) * 3x^2.

Logarithmic Functions

• Derivative Rule: Derivative of ln(u) is u'/u.
  • Example: Derivative of ln(x^2 + y^2) w.r.t X is 2x / (x^2 + y^2).

Complex Functions

• Quotient Rule: For differentiating f/g, use (g f' - f g') / g^2.
• Product Rule: For differentiating f * g, use f'g + fg'.*

Trigonometric Functions

• Derivative of Sin Function: Derivative of sin(u) is cos(u) * u'.
  • Example: For sin(x^3y^5), dz/dx is 3x^2y^5 * cos(x^3y^5).

Evaluating Partial Derivatives

• Evaluate at specific points by substituting the point values into the derived function.
• Example: For f(x, y) = 2x^3y^2 + 5y^3 + 4x^2, evaluate partial derivatives at (1, 2).

Higher Order Partial Derivatives

• Second Derivatives: Denoted by f_xx, f_xy, f_yx, f_yy.
  • Example: For f(x, y) = x^3 + 4x^5y^3 + 5y^4, find all second derivatives.

Mixed Partial Derivatives

• The order of taking derivatives does not affect the result if the function is continuous.
  • Example: f_xy equals f_yx.

Practical Examples

• Working through examples for various functions (polynomials, exponentials, logs, and trig functions).
• Demonstrating rules like product and quotient rules in context.

Summary

• Understanding rules and methods for finding partial derivatives, handling various functions and complexities.
• Proving mixed partial derivatives equality if functions are continuous.