Differentiation and Its Applications

Limits

Limits are fundamental in calculus, providing a precise way to describe the behavior of functions as their input values approach certain points. They are essential for defining derivatives and understanding continuity.

• Definition: The limit of a function f(x) as x approaches a is denoted by .

• Application: Used to analyze economic and financial metrics as they approach thresholds.

• Direct Substitution: For most functions, the limit can be found by substituting the value directly.

• Example:

• Non-Existence: Some limits do not exist, such as , which approaches infinity as x approaches 2.

Secant and Tangent Lines

The secant line connects two points on a curve, while the tangent line touches the curve at a single point. As the points get closer, the secant line approaches the tangent line.

• Slope of Secant:

• Limit Definition of Derivative:

Derivative: Definition and Notation

The derivative measures the rate of change of a function with respect to its variable. It is foundational in calculus and economics for analyzing marginal changes.

• Limit Definition:

• Notations: , , , ,

• Example: For ,

Derivatives of Common Functions

Several rules allow for efficient computation of derivatives for common functions.

• Example:

• Example:

• Example:

Rules of Differentiation

These rules simplify the process of finding derivatives for more complex functions.

• Constant Rule:

• Constant Multiple Rule:

• Sum/Difference Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Examples

• Product Rule:

• Quotient Rule:

• Chain Rule:

Higher Order Derivatives

Higher order derivatives are derivatives of derivatives, useful for analyzing concavity and inflection points.

• Example: For :

Critical Points and Classification

Critical points are where the derivative is zero or undefined. They are classified using the second derivative test.

• Steps:

  1. Find and .

  2. Set to find critical points.

  3. Use to classify:

     • : Maximum

     • : Minimum

     • : Further test needed (possible inflection point)

• Example: has critical points at (maximum) and (minimum).

Applications of Differentiation in Economics

Differentiation is used to analyze marginal changes in economic functions such as revenue, cost, and profit.

• Total Revenue (TR):

• Marginal Revenue (MR):

• Total Cost (TC):

• Marginal Cost (MC):

• Profit Maximization: Occurs when

• Profit ():

• Break-even Point: Where

• Elasticity of Demand:

Examples

• TR and MR: If , then ,

• MC: If , then ; at ,

• Profit Maximization: For , , maximum profit occurs at

• Monopolist Example: For , , maximum profit at , ,

Implicit Differentiation

Implicit differentiation is used when functions are defined implicitly, not explicitly solved for y in terms of x.

• Steps:

  1. Differentiate each term with respect to x.

  2. Treat y as a function of x; apply chain rule when differentiating y terms.

  3. Collect terms involving .

  4. Solve for .

• Example: For :

  • Collect terms:

  • Solve: