A student computes d/dx[sin(x)/cos(x)] using the quotient rule and gets −sec²(x) instead of sec²(x). What mistake did they make?
AThey forgot to square the denominator
BThey reversed the order in the numerator, computing fg′ − f′g instead of f′g − fg′
CThey differentiated cos(x) as +sin(x) instead of −sin(x)
DThey should have used the product rule instead of the quotient rule
With f = sin x and g = cos x, the correct numerator is f′g − fg′ = (cos x)(cos x) − (sin x)(−sin x) = cos²x + sin²x = 1. Reversing the order gives fg′ − f′g = (sin x)(−sin x) − (cos x)(cos x) = −sin²x − cos²x = −1, producing −sec²x. The order f′g − fg′ is non-negotiable — swapping it flips the sign of the entire result.
Question 2 Multiple Choice
Which of the following does NOT require the quotient rule to differentiate?
Ad/dx[sin(x)/cos(x)]
Bd/dx[(x³ + 1)/(x² − 1)]
Cd/dx[5x²/3]
Dd/dx[eˣ/(x + 1)]
When the denominator is a constant, simply factor it out: d/dx[5x²/3] = (5/3) · d/dx[x²] = (10/3)x. No quotient rule needed. The quotient rule is reserved for expressions where both numerator and denominator are genuine functions of x. Applying it unnecessarily to constant denominators is a common inefficiency. Options A, B, and D all have x-dependent denominators that require the rule.
Question 3 True / False
The quotient rule formula d/dx[f/g] = (f′g − fg′)/g² can be derived directly from the product rule applied to f(x) · [g(x)]⁻¹.
TTrue
FFalse
Answer: True
This derivation is the cleanest way to understand the quotient rule. Writing f/g as f · g⁻¹ and applying the product rule gives f′ · g⁻¹ + f · (−g′g⁻²). Simplifying: f′/g − fg′/g² = (f′g − fg′)/g². Knowing this derivation means you never need to memorize the formula blindly — you can reconstruct it from the product rule in under a minute.
Question 4 True / False
The numerator of the quotient rule result is f(x)g′(x) − f′(x)g(x) — 'hi d-lo minus lo d-hi.'
TTrue
FFalse
Answer: False
This is exactly backwards. The correct numerator is f′(x)g(x) − f(x)g′(x) — 'lo d-hi minus hi d-lo,' where 'hi' is the numerator f and 'lo' is the denominator g. The minus sign means order matters: reversing the two terms changes the sign of the entire result. Many students flip this and wonder why their answer is off by a sign.
Question 5 Short Answer
Explain why the quotient rule produces f′g − fg′ in the numerator and not fg′ − f′g. Where does this asymmetry come from?
Think about your answer, then reveal below.
Model answer: The asymmetry comes from the product rule applied to f · g⁻¹. The first term differentiates f (leaving g⁻¹ alone): f′/g. The second term differentiates g⁻¹ using the chain rule: f · (−g′/g²) = −fg′/g². Combining over a common denominator gives (f′g − fg′)/g². The minus sign is inherited from the chain rule applied to g⁻¹, and the order f′g first is fixed by which factor is differentiated first.
The non-commutativity is the key: f′g − fg′ ≠ fg′ − f′g (they differ by a sign). Because the formula comes from a minus sign in the chain rule, swapping the terms is not a cosmetic rearrangement — it produces the wrong sign entirely. This is why the mnemonic 'lo d-hi minus hi d-lo' specifies the order: d(hi) comes first, d(lo) is subtracted.