3.4.3 The Product and Quotient Rule

Problem Setup

• Objective: Differentiate a rational function that simultaneously requires the Product Rule for its numerator and the Quotient Rule for the overall fraction.

• Implicitly-defined function (from context):
  f(x)=\frac{4x\,e^x}{x^2+1}
  • Numerator: N(x)=4x\,e^x
  • Denominator: D(x)=x^2+1

Step 1 – Prepare for the Quotient Rule

• Reminder of the Quotient Rule: For f(x)=\dfrac{N(x)}{D(x)},
  f'(x)=\frac{N'(x)\,D(x)-N(x)\,D'(x)}{\bigl(D(x)\bigr)^2}

• First requirement: compute N'(x).

Step 2 – Differentiate the Numerator with the Product Rule

• Product Rule statement: If N(x)=u(x)\,v(x), then N'(x)=u'(x)\,v(x)+u(x)\,v'(x).

• Identify parts:
  • u(x)=4x\;\Rightarrow\;u'(x)=4
  • v(x)=e^x\;\Rightarrow\;v'(x)=e^x

• Apply the rule:
  \begin{aligned}
  N'(x)&=4\cdot e^x+4x\cdot e^x\
  &=4e^x+4x e^x
  \end{aligned}

• Decision point: postpone simplification until later to keep algebra transparent.

Step 3 – Apply the Quotient Rule

Insert the computed pieces into the quotient-rule template:

\boxed{f'(x)=\frac{\bigl(4e^x+4x e^x\bigr)(x^2+1)-\bigl(4x e^x\bigr)\cdot(2x)}{\bigl(x^2+1\bigr)^2}}

Step 4 – Expand & Collect Like Terms (Numerator Only)

1. Distribute 4e^x+4x e^x across (x^2+1):
   • 4e^x(x^2)=4x^2 e^x
   • 4e^x(1)=4e^x
   • 4x e^x(x^2)=4x^3 e^x
   • 4x e^x(1)=4x e^x

2. Distribute the second product: -\,(4x e^x)(2x)= -8x^2 e^x

3. Aggregate all five terms:
   4x^3 e^x+(4x^2 e^x-8x^2 e^x)+4x e^x+4e^x
   • Combine 4x^2 e^x-8x^2 e^x=-4x^2 e^x

4. Resulting unsimplified numerator:
   4x^3 e^x-4x^2 e^x+4x e^x+4e^x

Step 5 – Factor for Elegance & Insight

• Notice every term shares a common factor of 4e^x.

• Factor it out:
  \begin{aligned}
  &4e^x\bigl(x^3- x^2+ x+1\bigr)
  \end{aligned}

• Final compact derivative:
  \boxed{f'(x)=\frac{4e^x\left(x^3-x^2+x+1\right)}{\left(x^2+1\right)^2}}

Key Takeaways & Connections

• Layered Rules: Complicated expressions often decompose into nested applications of basic rules. Preparing sub-results (here, N'(x)) off to the side streamlines the process.

• Factor Recognition: After heavy algebra, common factors (here 4e^x) frequently emerge; factoring them:
  • Clarifies structure
  • Simplifies future work (e.g.
  • analyzing critical points
  • canceling terms if part of a larger expression).

• Exponentials & Polynomials: The derivative preserves e^x in every term (due to \tfrac{d}{dx}e^x=e^x). Such echoes help when checking work—if an e^x disappears entirely, an algebraic slip likely occurred.

• Efficiency Tip: Decide strategically when to simplify—too early may obscure pattern recognition, too late can explode algebra.

• Real-world relevance: Quotients of a polynomial and an exponential model growth processes constrained by saturation (denominator). Accurate derivatives gauge instantaneous change, crucial in control theory, pharmacokinetics, or any feedback-limited growth.

Ethical / Philosophical Reflection

• While derivative rules look mechanical, they represent a deep philosophical idea: we can understand complex change by decomposing it into simple, local interactions (products & ratios). This mirrors problem-solving beyond math—break big issues into digestible parts.

Final Formula Recap

f(x)=\frac{4x e^x}{x^2+1},\qquad f'(x)=\frac{4e^x\left(x^3-x^2+x+1\right)}{\bigl(x^2+1\bigr)^2}