Overview
This lecture covers how to find the derivative of a function using the limit definition, demonstrating step-by-step solutions for various common functions.
The Definition of the Derivative
- The derivative of ( f(x) ), written ( f'(x) ), is ( \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} ).
- This process is known as the limit process or the definition of the derivative.
Examples of Finding Derivatives
Linear Function: ( f(x) = 5x - 4 )
- ( f(x+h) = 5(x+h) - 4 ).
- After simplifying, ( f'(x) = 5 ).
Quadratic Function: ( f(x) = x^2 )
- ( f(x+h) = (x+h)^2 ).
- After expanding and simplifying, ( f'(x) = 2x ).
Reciprocal Function: ( f(x) = 1/x )
- ( f(x+h) = 1/(x+h) ).
- Using the common denominator and simplifying, ( f'(x) = -1/x^2 ).
Square Root Function: ( f(x) = \sqrt{x} )
- ( f(x+h) = \sqrt{x+h} ).
- Multiply numerator and denominator by the conjugate to eliminate the radical.
- After simplification, ( f'(x) = 1/(2\sqrt{x}) ).
Fraction with Square Root in Denominator: ( f(x) = 8/\sqrt{x} )
- ( f(x+h) = 8/\sqrt{x+h} ).
- Use common denominator and conjugate method to simplify.
- The derivative simplifies to ( f'(x) = -4/x^{3/2} ).
General Polynomial: ( f(x) = x^2 - 5x + 9 )
- ( f(x+h) = (x+h)^2 - 5(x+h) + 9 ).
- After expanding and simplifying, ( f'(x) = 2x - 5 ).
Key Terms & Definitions
- Derivative — Measures the rate at which a function changes at any point; denoted as ( f'(x) ).
- Limit Process — Approach where ( h ) tends to zero to find the derivative using the formula.
- Conjugate — An expression used to rationalize numerators/denominators with radicals, changing the sign between terms.
Action Items / Next Steps
- Practice finding derivatives for additional functions using the definition.
- Review the process of simplifying algebraic expressions and using conjugates for limits with radicals.