3.4.2 The Product and Quotient Rule

Tangent Line Using the Quotient Rule

• Goal: Find the equation of the tangent line to the curve $f(x)=\frac{x^2}{x^2-4}$ at the point $(3,2)$ and then sketch both the curve and its tangent.

Derivative (Quotient Rule)

• Remember quotient rule: $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$.
• Here, $u=x^2\;(u'=2x)$, $v=x^2-4\;(v'=2x)$.
• Substitute:
• Numerator: $2x(x^2-4)-x^2(2x)$.
• Expand: $2x^3-8x-2x^3-2x=-10x$.
• Denominator: $(x^2-4)^2$.
• Final derivative: $f'(x)=\frac{-10x}{(x^2-4)^2}$.

Slope at $x=3$

• Evaluate $f'(3)=\frac{-30}{(9-4)^2}=\frac{-30}{25}=-\frac65$.

Tangent-Line Equation (Point-Slope Form)

• Point $P(3,2)$, slope $m=-\frac65$.
• Equation: $y-2=-\frac65\,(x-3)$.
• Optional slope-intercept: $y=-\frac65x+\frac{28}{5}$.

Graphical Interpretation

• The curve has vertical asymptotes at $x=\pm2$ (denominator zero).
• The tangent line touches the curve only at $x=3$ and has negative slope as computed.

Generalized Power Rule (Any Real Exponent $n$)

• Classic power rule: $\frac{d}{dx}(x^n)=n x^{n-1}$ required positive integer $n$.
• After proving with product & quotient rules, it now holds for “any real number” $n$ (including 0, negatives, fractions, irrationals).
• Practical implication: you may rewrite radicals or quotients as powers with negative / fractional exponents, then differentiate directly.

Worked Examples

1. Rational Function With Negative Exponent

• Original: $f(x)=\frac{9}{x^5}$.
• Rewrite: $f(x)=9x^{-5}$.
• Derivative: $f'(x)=9(-5)x^{-6}=-45x^{-6}=\frac{-45}{x^6}.$

2. Large Quotient Simplified First

• Original: $f(t)=\frac{3t^{16}-4}{t^6}$.
• Divide term-by-term: $f(t)=3 t^{10}-4 t^{-6}.$
• Differentiate with power rule:
• $\frac{d}{dt}(3t^{10})=30t^9$.
• $\frac{d}{dt}(-4t^{-6})=(-4)(-6)t^{-7}=24t^{-7}$.
• Result: $f'(t)=30t^9+\frac{24}{t^7}.$

3. Product of a Radical and an Exponential

• Function: $g(z)=\sqrt[3]{z}\,e^{z}=z^{1/3} e^{z}.$
• Use product rule $(fg)' = f'g + fg'$.
• First term derivative: $\frac13 z^{-2/3}.$
• Second term derivative: $e^{z}$ (self-derivative).
• Combine:
$g'(z)=\frac13 z^{-2/3} e^{z}+z^{1/3}e^{z}=e^{z}\Big(\frac{1}{3z^{2/3}}+z^{1/3}\Big).$
• Optional single-fraction form: $g'(z)=\frac{e^{z}(1+3z)}{3z^{2/3}}.$

4. Advanced Quotient: Radical-Power Over Quadratic

• Function: $h(x)=\frac{3x^{5/2}}{2x^2+4}.$
• Quotient rule setup:
• $u=3x^{5/2},\;u'=\frac{15}{2}x^{3/2}.$
• $v=2x^2+4,\;v'=4x.$
• Derivative:
$h'(x)=\frac{\,u'v-u v'\,}{v^2}=\frac{\frac{15}{2}x^{3/2}(2x^2+4)-3x^{5/2}(4x)}{(2x^2+4)^2}.$
• Simplify numerator:
• First distribute: $15x^{7/2}+30x^{3/2}$.
• Second product: $-12x^{7/2}$.
• Combine like terms: $3x^{7/2}+30x^{3/2}$.
• Factor common term: $3x^{3/2}(x^2+10)$.
• Final form:
$h'(x)=\frac{3x^{3/2}(x^2+10)}{(2x^2+4)^2}.$

Key Takeaways & Connections

• Quotient rule often benefits from strategic simplification (canceling terms, factoring negatives) before plugging numbers.
• Converting quotients / radicals to negative or fractional exponents streamlines differentiation by allowing direct use of the power rule.
• Product-rule derivatives involving $e^{x}$ retain $e^{x}$ in the result; factor it out to simplify.
• All differentiation techniques—power, product, quotient—intermesh: proving the generalized power rule relies on product & quotient rules.
• Geometric interpretation: the derivative is the slope of the tangent line; in rational functions, asymptotes affect graph shape and tangent behavior.

Ethical / Practical Notes

• Checking for domain issues (e.g.
• Denominator zero ⇒ vertical asymptotes at $x=\pm2$.
• Fractional exponents $\frac13$ require $z\ge0$ if restricting to real numbers.)
• Always simplify only when algebraically legitimate (avoid dividing by expressions that could be zero).
• Graphing tools ( calculators, CAS, plotting software ) are invaluable for verifying tangent-line location and slope visually.