Product Rule

Concept Explanation

1. Geometric Intuition (The Expanding Rectangle)

Consider a rectangle whose area is $A(x) = u(x) \cdot v(x)$, where both side lengths $u(x)$ and $v(x)$ change as $x$ varies. When $x$ increases by a small increment $\Delta x$, the side lengths increase by $\Delta u$ and $\Delta v$. The total change in area is:

Dividing by $\Delta x$ and taking the limit as $\Delta x \to 0$: The corner sliver $\Delta u \Delta v$ shrinks to zero faster than the two rectangular strips $u \Delta v$ and $v \Delta u$.

2. Common Pitfall: The False Product Rule

A frequent mistake is assuming $\frac{d}{dx}[uv] = u' \cdot v'$. For instance, if $f(x) = x \cdot x = x^2$, the correct derivative is $2x$. But $u' \cdot v' = 1 \cdot 1 = 1 \neq 2x$. Always use the sum of two cross-terms!

3. The Component Dictionary Strategy

To avoid misplacing terms or algebraic sign errors in multi-step problems, build a 4-part component dictionary before assembling the derivative:

Then assemble: $f'(x) = u'v + uv'$.

4. Extended Product Rule for Three or More Factors

For three differentiable functions $f(x) = u(x) \cdot v(x) \cdot w(x)$:

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Engineering Applications

• Mechanical Engineering (Dynamic Power Transmission): Mechanical power is the product of torque and angular velocity: $P(t) = \tau(t) \cdot \omega(t)$. The time rate of change of power is $\frac{dP}{dt} = \tau'(t)\omega(t) + \tau(t)\omega'(t)$.
• Electrical Engineering (Instantaneous Power): Electrical power delivered to an AC load is $p(t) = v(t) \cdot i(t)$. The derivative $\frac{dp}{dt} = v'(t)i(t) + v(t)i'(t)$ measures transient power spikes.
• Entertainment Engineering (Vibrational Rigs): The displacement profile of a damped show platform $y(t) = (at + b)\sin(\omega t)$ combines polynomial motion with harmonic oscillations, requiring the product rule to determine velocity.

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Example Problems

Example 1: Algebraic and Trigonometric Product

Find the derivative of $f(x) = x^3 \sin(x)$.

Solution

1. Build the Component Dictionary:
   • $u = x^3 \implies u' = 3x^2$
   • $v = \sin(x) \implies v' = \cos(x)$
2. Apply the Product Rule ($u'v + uv'$):
   $$f'(x) = (3x^2)(\sin x) + (x^3)(\cos x)$$
3. Factor out common factors:
   $$f'(x) = x^2 (3\sin x + x\cos x)$$

Example 2: Product of Linear and Radical Terms

Find the derivative of $g(t) = (2t - 5)\sqrt{t^2 + 1}$ or $g(t) = (2t - 5)\sqrt{t}$.

Solution

For $g(t) = (2t - 5)t^{\frac{1}{2}}$:

1. Method 1: Product Rule:
   • $u = 2t - 5 \implies u' = 2$
   • $v = t^{\frac{1}{2}} \implies v' = \frac{1}{2}t^{-\frac{1}{2}}$
     $$g'(t) = 2(t^{\frac{1}{2}}) + (2t - 5)\left(\frac{1}{2}t^{-\frac{1}{2}}\right) = 2\sqrt{t} + \sqrt{t} - \frac{5}{2\sqrt{t}} = 3\sqrt{t} - \frac{5}{2\sqrt{t}}$$
2. Method 2: Pre-Processing (Distributing first):
   $$g(t) = 2t^{\frac{3}{2}} - 5t^{\frac{1}{2}} \implies g'(t) = 2\left(\frac{3}{2}t^{\frac{1}{2}}\right) - 5\left(\frac{1}{2}t^{-\frac{1}{2}}\right) = 3\sqrt{t} - \frac{5}{2\sqrt{t}}$$
   Both methods yield the identical result.

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Connections & References

• Parent Concepts: Derivatives
• Related Rules: Power Rule, Quotient Rule, Chain Rule
• Trigonometric Connections: Derivatives of Trigonometric Functions
• Course Sessions: Calc-1 Session 07