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.