Concepts of the Derivative

Definition and Interpretation

The derivative of a function f(x) at a given value x_0 measures the instantaneous rate of change of the function at that point. It is also interpreted as the slope of the tangent line to the graph of f(x) at (x_0, f(x_0)).

• Limit Definition of the Derivative at a Point:

• General Derivative Function:

• Steps to Compute the Derivative Using the Limit Definition:

  1. Evaluate f(x + h)

  2. Compute the difference f(x + h) - f(x) and simplify

  3. Form the quotient and simplify

  4. Compute the limit as h approaches 0

• Differentiation is the process of finding the derivative of a function.

• Graphical Interpretation: The derivative at a point gives the slope of the tangent line to the graph at that point. If the graph has a vertical asymptote (vertical tangent) or is not smooth (has a corner or cusp), the derivative does not exist at that point.

• If a function f(x) is not continuous at a point, then f(x) is not differentiable at that point. (However, a function can be continuous but not differentiable at a point, such as at a sharp corner.)

Finding the Equation of the Tangent Line

Procedure

To find the equation of the tangent line to the graph of f(x) at a given point (a, f(a)):

1. Compute the derivative f'(a) to find the slope at x = a.

2. Use the point-slope form of a line:

Example Problems:

• (a) For at the point

• (b) For at

• (c) For at

Example Solution (a):

• Compute

• At ,

• Equation of tangent:

Indicated Derivatives

Finding Derivatives of Given Functions

To find the derivative for a given function y in terms of x, apply the appropriate differentiation rules (power rule, quotient rule, etc.).

• (a)

• (b)

Example Solutions:

• (a)

• (b)

Summary Table: Differentiability and Continuity

The following table summarizes the relationship between continuity and differentiability at a point:

Key Terms

• Derivative: The instantaneous rate of change of a function at a point.

• Tangent Line: A straight line that touches a curve at a point and has the same slope as the curve at that point.

• Differentiable: A function is differentiable at a point if its derivative exists at that point.

• Continuous: A function is continuous at a point if there is no break, jump, or hole at that point.

Additional info:

• For more complex functions, use rules such as the product rule, quotient rule, and chain rule to compute derivatives.

• Graphical features such as sharp corners, cusps, or vertical tangents indicate points where the derivative does not exist.