Lecture Notes: Calculus - Derivatives as Functions

Introduction to Derivatives

• Derivative as a function: Continuation of the discussion from previous sections.
• Definition: Derivative ( f'(x) ) is defined as the limit as ( h ) approaches zero of the difference quotient, ( \frac{f(x+h) - f(x)}{h} ).

Example 1: Derivative of ( f(x) = \sqrt{2 - 2x} )

1. Substitute: ( f(x+h) = \sqrt{2 - 2(x+h)} ) and ( f(x) = \sqrt{2-2x} ).
2. Difference Quotient: ( \frac{\sqrt{2 - 2(x+h)} - \sqrt{2-2x}}{h} ).
3. Zero over zero: Direct substitution gives ( \frac{0}{0} ).
4. Conjugate multiplication:
   • Multiply numerator and denominator by conjugate.
   • Simplify expression.
5. Limit Solution:
   • After simplification, ( -2h ) remains in numerator.
   • ( h ) terms cancel out.
   • Final derivative: ( f'(x) = \frac{-1}{\sqrt{2-2x}} ).

Example 2: Derivative of ( f(x) = \frac{x}{x+1} )

1. Substitute: ( f(x+h) = \frac{x+h}{x+h+1} ) and ( f(x) = \frac{x}{x+1} ).
2. Difference Quotient: ( \frac{\frac{x+h}{x+h+1} - \frac{x}{x+1}}{h} ).
3. Combine Fractions: Use common denominator.
4. Simplification:
   • Simplify and cancel terms in the numerator.
   • ( h ) terms cancel out after substituting limits.
5. Limit Solution:
   • Calculate limit as ( h \to 0 ).
   • Final derivative: ( f'(x) = \frac{1}{(x+1)^2} ).

Differentiability and Continuity

• Concept: If a function is differentiable at ( x = c ), it is continuous at ( x = c ).
• Differentiability implies continuity, but not vice versa.
• Example: Absolute value function ( f(x) = |x| ):
  • Continuous at 0 but not differentiable due to differing limits from left and right.

Cases of Non-differentiability

1. Discontinuity:
   • Derivative does not exist at points of discontinuity.
2. Corners and Cusps:
   • Sharp changes in direction or points where two lines meet (e.g., absolute value function).
3. Vertical Tangents:
   • Infinite slope means the derivative does not exist.
4. Endpoints:
   • Two-sided limits are not possible, leading to non-existing derivatives.

Graph Analysis

• Identifying Non-differentiable Points:
  • Corners, discontinuities, and endpoints.
  • Example given on interval [(-2, 2)]:
    • Points of non-differentiability at: (-1, 0, 1, -2, 2).

Conclusion

• Derivatives help understand the behavior and characteristics of functions.
• Important to identify where derivatives do not exist to fully understand the function's graph and behavior.