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Quotient Rule Overview

Quotient Rule: Used to find the derivative of a function that is the quotient of two functions.
Formula: If ( f(x) = \frac{g(x)}{h(x)} ), then the derivative ( f' ) is given by:
- ( f'(x) = \frac{(h(x)g'(x) - g(x)h'(x))}{(h(x))^2} )
Key Steps:
- Low (h), derivative of the high (g). - High (g), derivative of the low (h), with a minus sign.
- All over the square of what's below (( (h(x))^2 )).

Example of Applying the Quotient Rule

Setting up the problem:
- Identify ( g(x) ) and ( h(x) ), and write down the derivatives ( g'(x) ) and ( h'(x) ).
- Example Setup: ( f(x) = \frac{3x^2 + 5}{x^2} )
- Derivatives: ( g'(x) = 6x ) and ( h'(x) = 2x )
- Write the formula: ( f'(x) = \frac{(x^2)(6x) - (3x^2 + 5)(2x)}{(x^2)^2} )
- Emphasize that this expression should not be simplified, focusing on getting it correctly set up first.

Common Mistakes in Using the Quotient Rule

Wrong Order in Numerator: Subtraction is not commutative!
- Example: ( a - b ) is not equal to ( b - a ).
Ensure to maintain the correct signs and terms when applying derivatives.

Chain Rule Overview

Chain Rule: For composite functions ( f(g(x)) ), find the derivative using:
- ( (f(g(x)))' = f'(g(x))g'(x) )
Importance of deconstructing into inside and outside functions.
Example:
- For ( f(x) = (6x^2 + 4x)^3 ):
  - Inside function: ( u = 6x^2 + 4x )
  - Outside function: ( v = u^3 )
- Derivatives needed:
  - Inside: ( u' = 12x + 4 )
  - Outside: ( v' = 3u^2 )
- Final derivative: ( f'(x) = 3(6x^2 + 4x)^2 imes (12x + 4) )

Tips for Practice

Always identify inside and outside functions clearly.
Write down the chain rule steps on the side as a reminder during practice.
Use parentheses generously to avoid mistakes with order of operations.

Conclusion

Both the Quotient Rule and Chain Rule are foundational tools in calculus for finding derivatives.
Practice is essential to reduce mistakes, particularly in function decomposition and applying these rules correctly.