Section 3.2 – Product Rule (Differentiation)

3.2 – Product Rule for Derivatives

Fundamental Rule
  • If f(x)f(x) and g(x)g(x) are differentiable, then
    rac{d}{dx}igl[f(x)g(x)igr] \= f(x)g'(x)+g(x)f'(x).
  • Mnemonic: “Left · dRight + Right · dLeft” (symbolically: L dR+R dLL\,dR + R\,dL).
Conceptual Highlights
  • Differentiation distributes over addition but not over multiplication, hence the need for the product rule.
  • For more than two factors, differentiate one factor at a time and sum the resulting terms (Leibniz’s generalisation).
Verification Example (Pure Power Product)
  • Function: h(x)=x3 x5=x8.h(x)=x^3\,x^5=x^8.
  • Direct power‐rule derivative: h′(x)=8x7.h'(x)=8x^7.
  • Product‐rule check:
    1. f(x)=x3  (f′(x)=3x2),g(x)=x5  (g′(x)=5x4).f(x)=x^3\;(f'(x)=3x^2),\quad g(x)=x^5\;(g'(x)=5x^4).
    2. h′(x)=x3 5x4+x5 3x2=8x7.h'(x)=x^3\,5x^4 + x^5\,3x^2 = 8x^7.
  • Confirms internal consistency.
Worked Example 1

Given y=x2(x3−2x).y = x^2\bigl(x^3-2x\bigr).

  1. Identify parts:
    • f(x)=x2,  f′(x)=2x;f(x)=x^2,\; f'(x)=2x;
    • g(x)=x3−2x,  g′(x)=3x2−2.g(x)=x^3-2x,\; g'(x)=3x^2-2.
  2. Apply product rule:

    \begin{aligned}
    y' &= x^2(3x^2-2) + (x^3-2x)(2x) \
    &= (3x^4-2x^2) + (2x^4-4x^2) \
    &= 5x^4 - 6x^2.
    \end{aligned}
  3. Factor (optional): y′=x2(5x2−6).y' = x^2(5x^2-6).
  4. Horizontal tangents (y′=0y'=0):
    • x=0x=0 or x=±65.x = \pm\sqrt{\tfrac{6}{5}}.
Worked Example 2

Given y=(3x2+4)(2x2+3).y=(3x^2+4)(2x^2+3).

  1. Parts:
    • f(x)=3x2+4,  f′(x)=6x;f(x)=3x^2+4,\; f'(x)=6x;
    • g(x)=2x2+3,  g′(x)=4x.g(x)=2x^2+3,\; g'(x)=4x.
  2. Differentiate:

    \begin{aligned}
    y' &= (3x^2+4)(4x) + (2x^2+3)(6x) \
    &= 12x^3 + 16x + 12x^3 + 18x \
    &= 24x^3 + 34x.
    \end{aligned}
  3. No further simplification required.
Finding Horizontal Tangent Lines (General Steps)
  1. Compute y′y' via the product rule.
  2. Set y′=0y'=0.
  3. Solve for all real xx values—each solution gives an xx‐coordinate where the tangent is horizontal.
Common Pitfalls & Tips
  • Missing a term: Always differentiate both factors—two terms, no exceptions.
  • Order mix-up: Stick to the mnemonic L dR+R dLL\,dR + R\,dL.
  • Algebra errors: Combine like terms carefully after differentiating.
  • Factoring first: Often simplifies solving y′=0y'=0 for horizontal tangents.
Quick Reference Formulae
  • Product rule: ddx[f(x)g(x)]=f(x)g′(x)+g(x)f′(x).\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x).
  • Power rule: ddx[xn]=nxn−1.\frac{d}{dx}[x^n] = nx^{n-1}.
  • Horizontal tangent condition: y′(x)=0.y'(x)=0.