The quotient rule is a powerful tool for finding derivatives of functions divided by other functions. It's essential for tackling rational functions and expressions involving division, making it a key player in your differentiation toolkit.

Mastering the quotient rule opens doors to more complex problems. By combining it with other rules like product and chain, you'll be able to differentiate a wide range of functions, expanding your calculus problem-solving abilities.

The Quotient Rule

Formula of quotient rule

• Finds derivative of a function that is the quotient of two other functions $f(x)$ and $g(x)$
• Quotient $\frac{f(x)}{g(x)}$ is differentiable everywhere $g(x) \neq 0$
• Formula: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$
  • $f'(x)$ is derivative of $f(x)$
  • $g'(x)$ is derivative of $g(x)$
• Derived using product rule and chain rule

Application to rational functions

• Rational functions written as quotient of two polynomial functions ($\frac{x^2 + 3x - 1}{2x - 5}$)
• Steps to differentiate rational function using quotient rule:
  1. Identify $f(x)$ (numerator) and $g(x)$ (denominator)
  2. Find $f'(x)$ and $g'(x)$ separately
  3. Apply quotient rule formula
• Also used for expressions involving division ($\frac{\sin(x)}{e^x}$, $\frac{x}{\sqrt{x + 1}}$)

Efficiency of quotient rule

• Most efficient when function is quotient of two differentiable functions
• Simplify using algebra before differentiating if possible ($\frac{x^2 - 1}{x + 1}$ simplified to $x - 1$)
• Product rule more appropriate for product of two functions
• Chain rule more appropriate for composition of functions

Combining with other rules

• Some functions require combination of differentiation rules
• Chain rule necessary when quotient rule nested inside another function
  • $f(x) = \sin\left(\frac{x^2}{x + 1}\right)$
    1. Use quotient rule to find derivative of $\frac{x^2}{x + 1}$
    2. Apply chain rule to differentiate $\sin(x)$
• Product rule necessary when quotient multiplied by another function
  • $g(x) = x^2 \cdot \frac{\cos(x)}{e^x}$
    1. Use quotient rule to find derivative of $\frac{\cos(x)}{e^x}$
    2. Apply product rule to multiply by $x^2$