Definition and Computation of the Derivative

Secant and Tangent Lines

The concept of the derivative is closely related to the slope of lines drawn on the graph of a function. The secant line connects two points on the curve, while the tangent line touches the curve at a single point and represents the instantaneous rate of change at that point.

• Secant Line Slope: The slope between two points and is given by:

• Tangent Line Slope: As , the secant line approaches the tangent line. The slope of the tangent line at is:

• Instantaneous Rate of Change: The slope of the tangent line is also the instantaneous rate of change of at .

Example: For , the slope of the tangent line at is computed using the difference quotient for decreasing values of , approaching as .

Formal Definition of the Derivative

The derivative of a function at a point is defined as:

• , provided the limit exists.

• If exists, is said to be differentiable at .

Examples of Derivative Computation

Polynomial Functions

• Example 1:

  • Find :

  • Find and : ,

  • Equation of tangent lines:

    • At :

    • At :

Rational Functions

• Example 2:

  • Find :

  • Find and : ,

Root Functions

• Example 3:

  • Find :

  • Find and : ,

Conditions for Differentiability

Discontinuity and Non-Differentiability

A function may fail to be differentiable at a point for several reasons:

• Discontinuity: If is discontinuous at , then does not exist.

• Vertical Tangent: If the tangent line is vertical at , the derivative is undefined.

• Corner: If the function has a sharp corner at , the derivative does not exist.

Example: The function is continuous everywhere but not differentiable at due to a corner.

Example: The function has a vertical tangent at , so is undefined.

Practice Problems

Polynomial and Rational Functions

• Problem 1:

  • Find :

  • Find :

  • Tangent line at :

• Problem 2:

  • Find :

  • Find :

  • Tangent line at :

• Problem 3:

  • Find :

  • Find :

  • Tangent line at :

• Problem 4:

  • Find :

  • Find and (if defined): is undefined (division by zero),

• Problem 5:

  • Find :

  • Find and (if defined): , is undefined (square root of zero is defined, so ; but for , undefined)

Graphical Analysis of Differentiability

For functions given by graphs, determine points of non-differentiability by identifying discontinuities, corners, or vertical tangents.

• At points where the graph has a jump, cusp, or vertical tangent, the function is not differentiable.

Summary Table: Reasons for Non-Differentiability

Additional info: The notes expand on the formal definition of the derivative, graphical interpretation, and conditions for differentiability, providing examples and practice problems relevant to Business Calculus students.