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Quotient Rule Overview

  • The quotient rule is used to find the derivative of a function that is the quotient of two other functions.

  • Formula for the Quotient Rule:

    • If ( h(x) = \frac{f(x)}{g(x)} ), then ( h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} )

Step-by-Step Example

  • Example: Find the derivative of ( h(x) = \frac{3x^2 + 5}{5x^2} ) using the quotient rule (do not simplify).

Setup

  • Identify Functions:

    • High = ( f(x) = 3x^2 + 5 )

    • Low = ( g(x) = 5x^2 )

Derivatives of Functions

  • Calculate Derivative of High:

    • ( f'(x) = 6x ) (applying power rule)

  • Calculate Derivative of Low:

    • ( g'(x) = 10x ) (applying power rule)

Apply Quotient Rule

  • Substitute into the formula:

    • ( h'(x) = \frac{(5x^2)(6x) - (3x^2 + 5)(10x)}{[5x^2]^2} )

Organization

  • Make sure to organize every part correctly:

    • Identify and keep track of ( g ), ( f ), ( f' ), and ( g' )

    • Correct placement of terms is crucial in subtraction operations (order matters).

Key Techniques

  • Avoid Mistakes:

    • Be careful with the signs during subtraction, it’s a common mistake to switch terms.

    • Derivatives calculated need to stay organized; clearly layout with primes and functions referenced.

Importance of Setup

  • Setting up the problem correctly is vital for obtaining the right solution.

    • All terms must be organized and accurately referenced.

Chain Rule Overview

  • The chain rule is necessary when differentiating a composition of functions.

  • Formula for Chain Rule:

    • If ( y = f(g(x)) ), then ( y' = f'(g(x))g'(x) )

Step-by-Step Chain Rule Process

  • Identify the inside and outside functions:

    • For example: ( f(g(x)) = (6x^2 + 4x)^3 )

      • Inside function: ( g(x) = 6x^2 + 4x )

      • Outside function: ( f(u) = u^3 )

Derivatives of Inside and Outside Functions

  • Calculate derivatives:

    • Inside: ( g'(x) = 12x + 4 )

    • Outside: ( f'(u) = 3u^2 )

Putting it All Together

  • Combine the derivatives:

    • Final result using chain rule --> ( y' = 3(6x^2 + 4x)^2 (12x + 4) )

Conclusion

  • Practice with both the quotient rule and chain rule is essential to reinforce understanding and application.

  • Knowing how to correctly layout and compute derivatives will help avoid common pitfalls in calculus.