The Product and Quotient Rules in Differentiation

The Product Rule

• The Product Rule is used to differentiate the product of two functions.

• Definition: The derivative of the product of two differentiable functions is equl to the first function times the derivative of the second plus the second function times the derivative of the first.

• Standard Formula: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$

• Example 2 Solution: For the function $y = (x^{-1} + 1)(x - 1)$, the Product Rule results in the derivative: $y' = \frac{x^2 + 1}{x^2}$

• The rule can be extended to products containing more than two differentiable factors (e.g., $f$, $g$, and $h$).

Differentiating Products vs. Constants

• Constant Multiple Rule: Used when one factor is a constant (e.g., $y = 2(2x+3)$).

• Product Rule: Used when both factors are variable quantities (e.g., $F(x) = f(x)g(x)$).

• Example 3 Comparison:

  • $y = (3x - 2x^2)(5 + 4x)$ requires the Product Rule as both are variables.

  • $y = 2(2x + 3)$ uses the Constant Multiple Rule: $y' = 2(2) = 4$.

The Quotient Rule

• The Quotient Rule allows for the differentiation of rational functions.

• The derivative of a quotient is NOT the quotient of the derivatives: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] \neq \frac{f'(x)}{g'(x)}$

• Standard Formula (derived from Example 8): $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

• Example 5: For the function $y = \frac{x - 1}{x + 1}$, the slope at point $(1, -1)$ is found by evaluating $\frac{dy}{dx}$ at $x = 1$.

Differentiation Efficiency and Alternatives

• The Quotient Rule is not always the most efficient method. If a quotient has a constant denominator, the Constant Multiple Rule is preferred.

• Example 7 Efficiency:

  • Original: $y = \frac{x^2 + 3x}{6}$

  • Rewrite: $y = \frac{1}{6}(x^2 + 3x)$

  • Differentiate: $y' = \frac{1}{6}(2x+3)$

• Rewriting quotients as products with negative exponents often simplifies calculations (e.g., rewriting $y = \frac{9}{5x^2}$ as $y = \frac{9}{5}x^{-2}$).

Application: Rate of Change in Blood Pressure

• Systolic blood pressure $P$ (in millimeters of mercury) over time $t$ (in seconds) is given by: $P = \frac{25t^2 + 125}{t^2 + 1}$

• The rate of change is found using the Quotient Rule: $\frac{dP}{dt} = \frac{(t^2 + 1)(50t) - (25t^2 + 125)(2t)}{(t^2 + 1)^2} = \frac{-200t}{(t^2 + 1)^2}$

• At $t = 5$ seconds, the pressure change rate is approximately $-1.48\,mm\,Hg/s$, indicating the pressure is dropping.