Derivatives

Definition and Fundamental Concept

The derivative of a function at a point measures the instantaneous rate of change of the function with respect to its variable. It is also interpreted as the slope of the tangent line to the graph at that point.

• Derivative at a Point: For a function f(x), the derivative at x = a is defined as:

• Derivative as a Function: The derivative function f'(x) gives the slope of the tangent at any point x:

• Interpretation: The derivative represents the rate of change and the slope of the tangent line to the curve y = f(x) at x.

Examples of Derivative Computation

• Example 1: For f(x) = 3x - 8, the derivative is:

• Example 2: For f(x) = x^2, the derivative is:

• Example 3: For f(x) = |x|, the derivative is not defined at x = 0 due to a corner.

Differentiability

Conditions for Differentiability

A function f(x) is differentiable at x = a if the derivative exists at that point. Differentiability implies continuity, but not all continuous functions are differentiable.

• Not Differentiable at x = a if:

  • There is a hole or discontinuity at x = a.

  • The function has a vertical tangent at x = a.

  • The function has a corner or cusp at x = a.

Summary Table: Differentiability Criteria

Graphical Interpretation of Derivatives

Relationship Between Derivative and Graph

The sign and value of the derivative at a point indicate the behavior of the function:

• f'(x) > 0: Function is increasing at x.

• f'(x) < 0: Function is decreasing at x.

• f'(x) = 0: Function has a horizontal tangent (possible local maximum or minimum).

Example: Sketching Derivative from Function Graph

• Given a graph of f(x), the derivative f'(x) can be sketched by noting where the function is increasing, decreasing, or constant.

• Points where f(x) has horizontal tangents correspond to zeros of f'(x).

• Vertical tangents, corners, or discontinuities in f(x) correspond to points where f'(x) does not exist.

Example Table: Behavior of f(x) and f'(x)

Summary

• The derivative measures the instantaneous rate of change and slope of the tangent line.

• Not all functions are differentiable everywhere; discontinuities, vertical tangents, and corners prevent differentiability.

• The derivative function f'(x) provides information about the increasing/decreasing behavior and critical points of f(x).

Additional info: Some context and examples were inferred from standard calculus curriculum to clarify fragmented notes and ensure completeness.