Lecture on Rules for Finding Derivatives

Overview

• Discussed key rules for finding derivatives in calculus.
• Focus on rules such as the derivative of a constant, power rule, product rule, and quotient rule.

Key Derivative Rules

1. Derivative of a Constant

• Rule: Derivative of a constant is always 0.
• Formula: ( \frac{d}{dx}(c) = 0 )
• Examples:
  • If ( y = 3 ), then ( \frac{dy}{dx} = 0 ).
  • If ( y = 1000 ), then ( \frac{dy}{dx} = 0 ).

2. Power Rule

• Rule: Useful for any power of ( x ).
• Formula: ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} )
• Examples:
  • ( y = x^3 ), ( \frac{dy}{dx} = 3x^2 )
  • ( y = x^5 ), ( \frac{dy}{dx} = 5x^4 )

3. Derivative of a Constant Times a Function

• Rule: Isolate the constant and take the derivative of the function.
• Formula: ( \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x)) )
• Examples:
  • ( y = 3x^6 ), ( \frac{dy}{dx} = 18x^5 )
  • ( y = 2x^2 ), ( \frac{dy}{dx} = 4x )

4. Derivative of Sum and Difference

• Rule: Derivative of a sum/difference is the sum/difference of derivatives.
• Formula: ( \frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx} )
• Examples:
  • ( y = 3x^2 + 2x - 1 ), ( \frac{dy}{dx} = 6x + 2 )
  • ( y = \frac{2}{3}x^3 - 4x + 86 ), ( \frac{dy}{dx} = 2x^2 - 4 )

5. Product Rule

• Rule: Used for products of two functions.
• Formula: ( \frac{d}{dx}(u \cdot v) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} )
• Examples:
  • ( y = (x+1)(x+2) ), ( \frac{dy}{dx} = 2x + 3 )
  • ( y = (x^3 + 2x)(2x - 1) ), ( \frac{dy}{dx} = 8x^3 - 3x^2 + 8x - 2 )
  • ( y = (x^3 - 3x^2)(x^2 + 4x + 2) ), apply product rule and simplify.

6. Quotient Rule

• Rule: Used for quotients of two functions.
• Formula: ( \frac{d}{dx}(\frac{u}{v}) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} )
• Examples:
  • ( y = \frac{x}{x+1} ), ( \frac{dy}{dx} = \frac{1}{(x+1)^2} )
  • ( y = \frac{2x}{x^2+1} ), ( \frac{dy}{dx} = \frac{2 - 2x^2}{(x^2+1)^2} )

Summary

• Derivatives measure the rate of change of functions.
• Understanding these basic rules is essential for solving more complex calculus problems.
• Practice applying these rules to different functions for mastery.