Lecture Notes: Practice with the Difference Quotient

Introduction

• Difference Quotient: A formula used to define the slope of a function as a function of x.
• Formula: ( \frac{f(x + h) - f(x)}{h} )
• Important to memorize this formula for computing derivatives.

Practice with Different Functions

1. Polynomial Function Example

• Process:
  • Replace every instance of x in ( f(x) ) with ( x + h ).
  • Example: ( (x + h)^2 + (x + h) ).
  • Simplify by expanding and canceling terms.
• Steps:
  • Expand ((x + h)^2) to (x^2 + 2xh + h^2).
  • Distribute negative in (- (x^2 + x)).
  • Simplify: Cancel (x^2) terms, simplify remaining terms.
  • Factor (h) out: ( \frac{2xh + h^2 + h}{h} \rightarrow 2x + h + 1 ), for (h \neq 0).

2. Root Function Example

• Difference Quotient for function g: ( g(x + h) - g(x) ) over (h).
• Steps:
  • Replace x with (x + h) under the square root.
  • Rationalize the numerator by multiplying by the conjugate.
  • Use difference of squares: ((a-b)(a+b) = a^2 - b^2).
  • Simplify: Cancel (h) terms, ensure (h \neq 0).
• Result: Shift roots from numerator to denominator, useful for derivatives.

3. Rational Function Example

• Difference Quotient: ( \frac{1}{x + h - 2} - \frac{1}{x - 2} ) over (h).
• Steps:
  • Multiply by least common denominator to eliminate complex fractions.
  • Simplify by distributing and canceling terms.
  • Simplify: Cancel (h), ensure (h \neq 0).
• Result: ( \frac{-1}{(x+h-2)(x-2)} ).

Conclusion

• These are foundational methods for computing derivatives using the difference quotient.
• Necessary skills for upcoming discussions and more complex calculus problems.