Lecture Notes: Finding Derivatives Using the Definition of the Derivative

Introduction to Derivatives

• Definition of Derivative Formula:
  • ( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} )
  • Purpose: To find the derivative of a function using the limit process.

Example 1: Linear Function

• Function: ( f(x) = 5x - 4 )
• First Derivative:
  1. Substitute ( f(x+h) = 5(x+h) - 4 ) into the derivative formula.
  2. Simplify: ( \lim_{{h \to 0}} \frac{5x + 5h - 4 - (5x - 4)}{h} )
  3. Simplification leads to: ( \frac{5h}{h} = 5 )
• Result: ( f'(x) = 5 )

Example 2: Quadratic Function

• Function: ( f(x) = x^2 )
• First Derivative:
  1. Substitute ( f(x+h) = (x+h)^2 ) into the derivative formula.
  2. Simplify: ( \lim_{{h \to 0}} \frac{x^2 + 2xh + h^2 - x^2}{h} )
  3. Combine like terms: ( \lim_{{h \to 0}} \frac{2xh + h^2}{h} )
  4. Cancel ( h ) terms: ( \lim_{{h \to 0}} (2x + h) )
  5. Substitute: ( 2x )
• Result: ( f'(x) = 2x )

Example 3: Rational Function

• Function: ( f(x) = \frac{1}{x} )
• First Derivative:
  1. Substitute ( f(x+h) = \frac{1}{x+h} ) into the derivative formula.
  2. Simplify using common denominator: ( \lim_{{h \to 0}} \frac{x - (x+h)}{h(x)(x+h)} )
  3. Cancel ( h ) terms: ( \lim_{{h \to 0}} \frac{-1}{x(x+h)} )
  4. Substitute: ( \frac{-1}{x^2} )
• Result: ( f'(x) = \frac{-1}{x^2} )

Example 4: Square Root Function

• Function: ( f(x) = \sqrt{x} )
• First Derivative:
  1. Substitute ( f(x+h) = \sqrt{x+h} ) into the derivative formula.
  2. Multiply by conjugate: ( \lim_{{h \to 0}} \frac{x - (x+h)}{h(\sqrt{x} + \sqrt{x+h})} )
  3. Cancel ( h ) terms: ( \lim_{{h \to 0}} \frac{1}{\sqrt{x} + \sqrt{x+h}} )
  4. Substitute: ( \frac{1}{2\sqrt{x}} )
• Result: ( f'(x) = \frac{1}{2\sqrt{x}} )

Example 5: Complex Fraction with Square Root

• Function: ( f(x) = \frac{8}{\sqrt{x}} )
• First Derivative:
  1. Substitute ( f(x+h) = \frac{8}{\sqrt{x+h}} ) into the derivative formula.
  2. Simplify using conjugate and combine fractions.
  3. Cancel ( h ) terms and simplify.
  4. Substitute to find derivative.
• Result: ( f'(x) = \frac{-4}{x\sqrt{x}} )

Example 6: Polynomial Function

• Function: ( f(x) = x^2 - 5x + 9 )
• First Derivative:
  1. Substitute ( f(x+h) = (x+h)^2 - 5(x+h) + 9 ) into the derivative formula.
  2. Simplify: ( \lim_{{h \to 0}} \frac{h^2 + 2xh - 5h}{h} )
  3. Cancel ( h ) terms and simplify.
  4. Substitute: ( 2x - 5 )
• Result: ( f'(x) = 2x - 5 )

Conclusion

• Understanding how to find derivatives using the limit process is fundamental for calculus.
• Practice using the definition and simplifying expressions to become proficient in calculating derivatives.