Quotient Rule for differentiation

Quotient Rule:

Rule:

  • For two differentiable functions u(x) and v(x) where v(x)0, the derivative of their quotient is: ddx[u(x)v(x)]=v(x)u(x)u(x)v(x)[v(x)]2.

Explanation:

  • The quotient rule states that when differentiating the quotient of two functions, the derivative is found using a specific formula involving the derivatives of the numerator and denominator.

Example:

  • Consider f(x)=x2sin(x)
    • Let u(x)=x2 and v(x)=sin(x)
    • u(x)=2x and v(x)=cos(x)
    • Apply the quotient rule: ddx[x2sin(x)]=sin(x)2xx2cos(x)[sin(x)]2

Properties and Notes:

  1. Applicability and Conditions:

    • The quotient rule is applicable when differentiating a quotient of two functions, and the denominator function is not zero.
  2. Avoiding Ambiguity:

    • The rule helps avoid the ambiguity that arises when directly differentiating fractions using the power rule.
  3. Handling Complex Ratios:

    • Useful in finding derivatives of complex algebraic expressions or functions expressed as ratios.

Applications of the Quotient Rule:

  • Rate of Change Problems:

    • Used to analyze rates of change in various contexts, such as motion problems in physics or growth and decay problems in biology or economics.
  • Optimization and Critical Points:

    • Helpful in optimization problems where finding critical points or extreme values of functions is necessary.