1000 Hilltop Cir 2

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

1. 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.