Questions: Product Rule

Question 1

A student differentiates h(x) = x²sin(x) by reasoning: 'The derivative of x² is 2x and the derivative of sin(x) is cos(x), so h'(x) = 2x·cos(x).' What error did they make?

• A. They should have used the chain rule instead of the product rule
• B. They multiplied the two derivatives together instead of applying the product rule
• C. They forgot to add a constant of integration
• D. They differentiated sin(x) incorrectly

Answer: (B)

The student committed the most common calculus error: assuming (fg)' = f'g'. The correct product rule gives h'(x) = 2x·sin(x) + x²·cos(x) — differentiate each factor while holding the other fixed, then add both results. Multiplying the derivatives (2x·cos(x)) gives the wrong answer because it misses the cross-term interactions.

Question 2

Which expression correctly gives d/dx[x³·eˣ]?

• A. 3x²·eˣ + x³·eˣ
• B. 3x²·eˣ
• C. x³·eˣ
• D. 3x²·eˣ · x³·eˣ

Answer: (A)

Applying the product rule with f = x³ and g = eˣ: f' = 3x² and g' = eˣ. The result is f'g + fg' = 3x²·eˣ + x³·eˣ. Option 1 is the 'just differentiate the first factor' error; option 2 is the 'just differentiate the second factor' error; option 3 multiplies the derivatives together — all three reflect the mistaken belief that (fg)' = f'g'.

Question 3

The product rule is needed whenever you differentiate an expression involving multiplication.

• T. True
• F. False

Answer: False

If one factor is a constant, the constant multiple rule suffices — d/dx[5sin(x)] = 5cos(x) directly, no product rule needed. The product rule is only required when BOTH factors depend on x. Recognizing when to use the simpler rule (constant multiple) vs. the product rule is itself a key skill.

Question 4

The product rule correctly states that d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

• T. True
• F. False

Answer: True

This is the product rule. The geometric intuition: imagine a rectangle with sides f and g (area = fg). When x increases slightly, the area gains two strips — one of height g and width f'Δx, one of width f and height g'Δx — giving (f'g + fg')Δx in total new area. The tiny corner piece (f'g'(Δx)²) vanishes in the limit, leaving exactly this formula.

Question 5

Why is the derivative of a product NOT simply the product of the derivatives? Explain using an example.

Model answer: When two quantities both depend on x, each one's change interacts with the other's current value — creating two cross-terms, not one. For example, if f(x) = g(x) = x, then f(x)g(x) = x², whose derivative is 2x. But f'(x)·g'(x) = 1·1 = 1 ≠ 2x. The product rule captures both interactions: f'g = 1·x = x and fg' = x·1 = x, summing to 2x.

The error (fg)' = f'g' treats the two factors as independent, but they are not — they share the same variable x. The rectangle analogy makes this concrete: two sides both growing simultaneously create two strips of new area, not one. Calculus must account for both.