Lecture Notes: Rules for Finding the Derivative

1. Derivative of a Constant

• Rule: The derivative of any constant is always zero.
  • Example: If ( y = 3 ), then ( \frac{dy}{dx} = 0 ).
  • Formula: ( \frac{d}{dx}(C) = 0 )

2. Power Rule

• Rule: ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} )
  • Example 1: ( y = x^3 \rightarrow \frac{dy}{dx} = 3x^2 )
  • Example 2: ( y = x^5 \rightarrow \frac{dy}{dx} = 5x^4 )

3. Derivative of a Constant Times a Function

• Rule: ( \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)) )
  • Example: If ( y = 3x^6 ), then ( \frac{dy}{dx} = 18x^5 )

4. Derivative of a Sum and Difference

• Rule: ( \frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx} )
  • Example: If ( y = 3x^2 + 2x - 1 ), then ( \frac{dy}{dx} = 6x + 2 )

5. Derivative of a Product

• Formula: ( \frac{d}{dx}(uv) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} )
  • Example: If ( y = (x+1)(x+2) ), then ( \frac{dy}{dx} = 2x + 3 )

6. Derivative of a Quotient

• Formula: ( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} )
  • Example: If ( y = \frac{x}{x+1} ), then ( \frac{dy}{dx} = \frac{1}{(x+1)^2} )

Additional Concepts

• Using FOIL Method: Useful in simplifying expressions before taking derivatives.
• Combining Like Terms: Important for simplifying results after applying derivative rules.

Tips

• Memorize the formulas and rules for quick application.
• Simplify expressions whenever possible to avoid complex calculations.
• Use mnemonic devices like "low D high minus high D low over low squared" for the quotient rule.

Practice

• Apply these rules to different functions to strengthen understanding.
• Try converting complex functions into simpler ones using algebraic manipulation before differentiation.

Thank you for attending the lecture on derivatives. Remember to practice the rules, and don't hesitate to use them on various problems to improve your calculus skills.