Overview
This lecture explains how to find the derivative of a function given as a quotient (division) of two functions, using the quotient rule, and provides a detailed example plus a practice problem.
The Quotient Rule for Derivatives
- The derivative of a quotient ( f(x) = \frac{g(x)}{h(x)} ) is found using a specific formula.
- The quotient rule states: ( f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} ).
- Always start by identifying the numerator ( g(x) ) and denominator ( h(x) ).
- Compute the derivative of the numerator and denominator separately before applying the rule.
Worked Example
- Given ( f(x) = \frac{2x - 3}{x^2} ), find derivatives: numerator ( g'(x) = 2 ), denominator ( h'(x) = 2x ).
- Apply the quotient rule: ( f'(x) = \frac{x^2 \cdot 2 - (2x-3) \cdot 2x}{(x^2)^2} ).
- Expand and simplify the numerator: distribute and combine like terms carefully, minding negative signs.
- Further simplify by factoring out common terms and reducing powers, e.g., dividing numerator and denominator by ( x ) where possible.
- Final answer: ( f'(x) = \frac{-2x + 6}{x^3} ).
Practice Exercise Provided
- Suggested exercise: find the derivative of ( \frac{3x^2 + 5}{2x^3} ).
- Steps: find derivatives ( g'(x) = 6x ), ( h'(x) = 6x^2 ), apply quotient rule, expand, factor, and simplify.
- Remember to only cancel terms when they are factors, not individual terms in sums or differences.
Key Terms & Definitions
- Quotient Rule — A formula for differentiating functions divided by each other: ( \frac{g}{h}' = \frac{h g' - g h'}{h^2} ).
- Numerator (g(x)) — The function on top of the fraction.
- Denominator (h(x)) — The function on the bottom of the fraction.
- Derivative — The rate at which a function changes with respect to its variable.
Action Items / Next Steps
- Solve the given practice problem: find the derivative of ( \frac{3x^2 + 5}{2x^3} ) using the quotient rule.
- Review previous videos if you are new to derivatives fundamentals.
- Prepare for future lessons on the chain rule and additional derivative techniques.