Quotient Rule Calculator — Derivative of f/g with Steps

Calculus Tool

Quotient Rule Calculator

Q: What is the quotient rule?

The quotient rule states that the derivative of a fraction f(x)/g(x) is: d/dx[f/g] = (f'g − fg') / g². Memory mnemonic: 'Low d-High minus High d-Low, square the bottom and away we go.' Low = g (denominator), High = f (numerator).

Q: What is the formula for the quotient rule?

The quotient rule formula: d/dx[f/g] = (f'g − fg') / g². Equivalent forms: (f/g)' = (f'g − fg') / g², h'(x) = [f'(x)g(x) − f(x)g'(x)] / [g(x)]². The order of subtraction matters — f'g comes first, fg' is subtracted. Reversing changes the sign of the entire result.

Q: What is the 'Low d-High minus High d-Low' mnemonic?

'Low d-High minus High d-Low, square the bottom and away we go' remembers d/dx[f/g] = (f'g − fg') / g². Low = g (denominator), High = f (numerator), d-High = f', d-Low = g'.

Q: What is the difference between the quotient rule and the product rule?

Product rule: d/dx[f·g] = f'g + fg' — for two functions multiplied. Quotient rule: d/dx[f/g] = (f'g − fg') / g² — for two functions divided. The quotient rule is derived from the product rule by writing f/g = f·g⁻¹ and applying product rule + chain rule.

Q: How do you derive tan(x) using the quotient rule?

tan(x) = sin(x)/cos(x). Apply quotient rule with f=sin(x), g=cos(x), f'=cos(x), g'=−sin(x). Numerator: cos²(x) + sin²(x) = 1 (Pythagorean identity). Denominator: cos²(x). Result: 1/cos²(x) = sec²(x).

Q: When should you use the quotient rule vs rewriting as a product?

Both approaches give the same answer. Use the quotient rule directly when the fraction form is clear. Rewrite as f·g⁻¹ and use the product rule when that simplifies the algebra. For constant denominators, just divide the derivative — no quotient rule needed.

Q: How do you apply the quotient rule with the chain rule?

When the numerator f or denominator g is a composite function, apply the chain rule to find f' or g' first, then substitute into the quotient rule (f'g − fg') / g². Chain rule comes before the quotient rule assembly.

Apply the quotient rule d/dx[f/g] = (f'g − fg')/g² to any fraction of functions — color-coded f (crimson) and g (amber), proper fraction display, trig derivations, and full step-by-step working.

d/dx [

] =

f'·g − f·g'

g²

Quotient Rule Calculator — d/dx[f/g] = (f'g − fg') / g²

Enter full fraction (e.g. sin(x)/x):

Use / for division, ^ for powers. Auto-detects f and g by splitting on top-level /.

d/dx[f/g] = (f'g−fg')/g² = ?

d/d

sin(x)/x

(x²+1)/(x−1)

eˣ/cos(x)

ln(x)/x²

(x³−1)/(x+1)

x/eˣ

sin/cos → tan

1/cos → sec

Quotient Rule Result — d/dx[f/g] = (f'g − fg') / g²

Step-by-Step Working

"Low d-High minus High d-Low, square the bottom and away we go" = (Low·dHigh − High·dLow) / Low² — this is (f'g − fg') / g²

All four trig derivatives below are derived directly from the quotient rule (f'g − fg') / g² using sin/cos definitions. Click "Try in Calculator" on any card to compute it instantly.

tan(x)

Definition: sin(x) / cos(x)

d/dx[tan(x)] =

cos(x)·cos(x) − sin(x)·(−sin(x))

cos²(x)

f = sin(x), g = cos(x)

f' = cos(x) | g' = −sin(x) ← note minus sign!

Numerator: cos²(x) − sin(x)·(−sin(x)) = cos²(x) + sin²(x)

Pythagorean identity: cos²(x) + sin²(x) = 1

Result: 1 / cos²(x) = sec²(x)

d/dx[tan(x)] = sec²(x)

Verified ✓

cot(x)

Definition: cos(x) / sin(x)

d/dx[cot(x)] =

(−sin(x))·sin(x) − cos(x)·cos(x)

sin²(x)

f = cos(x), g = sin(x)

f' = −sin(x) | g' = cos(x)

Numerator: −sin²(x) − cos²(x) = −(sin²+cos²) = −1

Result: −1 / sin²(x) = −csc²(x)

d/dx[cot(x)] = −csc²(x)

Verified ✓

sec(x)

Definition: 1 / cos(x)

d/dx[sec(x)] =

0·cos(x) − 1·(−sin(x))

cos²(x)

f = 1, g = cos(x)

f' = 0 | g' = −sin(x)

Numerator: 0·cos(x) − 1·(−sin(x)) = sin(x)

Result: sin(x)/cos²(x) = (1/cos(x))·(sin(x)/cos(x))

= sec(x)·tan(x)

d/dx[sec(x)] = sec(x)·tan(x)

Verified ✓

csc(x)

Definition: 1 / sin(x)

d/dx[csc(x)] =

0·sin(x) − 1·cos(x)

sin²(x)

f = 1, g = sin(x)

f' = 0 | g' = cos(x)

Numerator: 0·sin(x) − 1·cos(x) = −cos(x)

Result: −cos(x)/sin²(x) = −(1/sin(x))·(cos(x)/sin(x))

= −csc(x)·cot(x)

d/dx[csc(x)] = −csc(x)·cot(x)

Verified ✓

Function	Definition (quotient)	Derivative via quotient rule
tan(x)	sin(x)/cos(x)	sec²(x)
cot(x)	cos(x)/sin(x)	−csc²(x)
sec(x)	1/cos(x)	sec(x)·tan(x)
csc(x)	1/sin(x)	−csc(x)·cot(x)

When numerator f or denominator g is composite, apply the chain rule first, then substitute into the quotient rule (f'g − fg') / g².

f(x) — Numerator (may be composite):

f =

g(x) — Denominator (may be composite):

g =

d/d

sin(x²)/(x+1)²

e^(2x)/ln(x)

√(x+1)/(x²−1)

cos(x²)/e^(x²)

Quotient + Chain Rule Result

Step-by-Step Working

∂/∂x[f/g] = (∂f/∂x·g − f·∂g/∂x) / g² Same formula — treat all other variables as constants

f — Numerator:

f =

g — Denominator:

g =

Differentiate ∂/∂

x²y / (x+y)

eˣy / (xy+1)

sin(xy) / x²

Partial Derivative Quotient Rule Result

Step-by-Step Working

The Quotient Rule Formula — (f'g − fg') / g²

This quotient rule calculator applies the quotient rule d/dx[f/g] = (f'g − fg')/g² to any fraction of functions — with color-coded f (crimson, numerator) and g (amber, denominator), proper fraction rendering, and automatic verification.

The quotient rule states: if h(x) = f(x)/g(x), then h'(x) = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]². The order matters — f'g first, minus fg' second. The formula (f'g − fg') / g² is one of the four essential derivative rules.

d/dx[f/g] = (f'·g − f·g') / g² "Low d-High minus High d-Low, square the bottom and away we go"

(f/g)' = (f'g−fg')/g²

Prime notation

d/dx[f/g]=(f'g−fg')/g²

Leibniz notation

h'=(f'g−fg')/g²

Function notation

"Low d-High minus High d-Low, square the bottom and away we go" — the classic quotient rule mnemonic. Low = g (denominator), High = f (numerator). Low·d(High) − High·d(Low) = f'g − fg', divided by Low² = g². The formula (f'g − fg') / g² always gives two terms in the numerator.

Quotient Rule Derived from Product Rule

The quotient rule (f'g − fg') / g² is derived from the product rule by writing f/g = f·g⁻¹:

d/dx[f·g⁻¹] = f'·g⁻¹ + f·d/dx[g⁻¹]
Chain rule on g⁻¹: d/dx[g⁻¹] = −g⁻²·g'
Substitute: f'/g − fg'/g² = (f'g − fg') / g² ✓

How to Use the Quotient Rule — Step-by-Step Method

Step 1 — Identify f (numerator, "High") and g (denominator, "Low").
Step 2 — Find f'.
Step 3 — Find g'. If composite, apply chain rule.
Step 4 — Substitute into (f'g − fg') / g². f'g FIRST, then subtract fg'.
Step 5 — Simplify.

Example 1: d/dx[sin(x)/x]

f = sin(x), g = x; f' = cos(x), g' = 1
Quotient rule (f'g − fg') / g²: (cos(x)·x − sin(x)·1) / x²
Answer: (x·cos(x) − sin(x)) / x²

Example 2: d/dx[(x²+1)/(x−1)]

f = x²+1, g = x−1; f' = 2x, g' = 1
Numerator: 2x(x−1) − (x²+1)·1 = 2x²−2x−x²−1 = x²−2x−1
Answer: (x²−2x−1) / (x−1)²

Example 3: d/dx[eˣ/cos(x)]

f = eˣ, g = cos(x); f' = eˣ, g' = −sin(x)
Numerator: eˣ·cos(x) − eˣ·(−sin(x)) = eˣ(cos(x)+sin(x))
Answer: eˣ(cos(x)+sin(x)) / cos²(x)

Quotient Rule Practice Problems — 8 Solved Examples

Eight fully worked quotient rule practice problems for use as a quotient rule worksheet. Each applies the formula (f'g − fg') / g² explicitly. Click to expand.

1 d/dx[x / sin(x)] = ?

f = x, g = sin(x); f' = 1, g' = cos(x)
Quotient rule (f'g − fg') / g²: (1·sin(x) − x·cos(x)) / sin²(x)
Answer: (sin(x) − x·cos(x)) / sin²(x)

2 d/dx[(x²+1) / (2x−1)] = ?

f = x²+1, g = 2x−1; f' = 2x, g' = 2
Numerator: 2x(2x−1) − (x²+1)·2 = 4x²−2x−2x²−2 = 2x²−2x−2
Answer: 2(x²−x−1) / (2x−1)²

3 d/dx[eˣ / x²] = ?

f = eˣ, g = x²; f' = eˣ, g' = 2x
Numerator: eˣ·x² − eˣ·2x = eˣ·x(x−2)
Answer: eˣ(x−2) / x³

4 d/dx[ln(x) / (x+1)] = ?

f = ln(x), g = x+1; f' = 1/x, g' = 1
Numerator: (1/x)(x+1) − ln(x)·1 = (x+1)/x − ln(x)
Answer: (x+1−x·ln(x)) / (x(x+1)²)

5 d/dx[sin(x)/cos(x)] = ? → derive tan(x) via quotient rule

f = sin(x), g = cos(x); f' = cos(x), g' = −sin(x)
Numerator via (f'g − fg') / g²: cos²(x) − sin(x)·(−sin(x)) = cos²(x)+sin²(x) = 1
Answer: 1/cos²(x) = sec²(x) ✓ — this proves d/dx[tan(x)] = sec²(x)

6 d/dx[1/cos(x)] = ? → derive sec(x) via quotient rule

f = 1, g = cos(x); f' = 0, g' = −sin(x)
Quotient rule (f'g − fg') / g²: (0·cos(x) − 1·(−sin(x))) / cos²(x) = sin(x)/cos²(x)
Answer: sec(x)·tan(x) ✓

7 d/dx[(x³−2x) / (x²+1)] = ?

f = x³−2x, g = x²+1; f' = 3x²−2, g' = 2x
Numerator: (3x²−2)(x²+1) − (x³−2x)·2x
Expand: 3x⁴+3x²−2x²−2 − 2x⁴+4x² = x⁴+5x²−2
Answer: (x⁴+5x²−2) / (x²+1)²

8 d/dx[e^(2x) / (x²+4)] = ? (chain rule in numerator)

f = e^(2x) — composite → f' = 2e^(2x)
g = x²+4, g' = 2x
Quotient rule (f'g − fg') / g²: (2e^(2x)(x²+4) − e^(2x)·2x) / (x²+4)²
Factor 2e^(2x): 2e^(2x)(x²−x+4) / (x²+4)²
Answer: 2e^(2x)(x²−x+4) / (x²+4)²

Deriving Trig Derivatives Using the Quotient Rule

All four trig derivatives are derived from the quotient rule (f'g − fg') / g² — see the interactive cards in Tab 2 above for full derivations with fraction display. The key insight for tan(x) is that the Pythagorean identity cos²(x)+sin²(x)=1 simplifies the numerator to 1, giving 1/cos²(x) = sec²(x).

Function	Definition	d/dx from quotient rule (f'g − fg') / g²
tan(x)	sin(x)/cos(x)	sec²(x)
cot(x)	cos(x)/sin(x)	−csc²(x)
sec(x)	1/cos(x)	sec(x)·tan(x)
csc(x)	1/sin(x)	−csc(x)·cot(x)

Quotient Rule vs Product Rule — Which to Use?

Both give the same answer for f/g. The quotient rule (f'g − fg') / g² is usually faster when the fraction form is clear. Writing f/g = f·g⁻¹ and using the product rule also works — it gives identical results. This addresses "product quotient and chain rule" searches: all three rules are connected.

Decision Guide

Clear fraction with distinct numerator/denominator?

→

Quotient Rule
(f'g − fg') / g²

Denominator is a simple negative power? e.g. x⁻²

→

Product Rule
d/dx[f·g⁻¹]

Denominator is a constant? e.g. f(x)/5

→

Constant Multiple
d/dx[f/5] = f'/5

Quotient Rule with Chain Rule — Composite Numerators and Denominators

When f or g is composite, apply the chain rule to get its derivative first, then substitute into the quotient rule (f'g − fg') / g².

Example: d/dx[sin(x²) / (x+1)²]

f = sin(x²) — chain rule: f' = 2x·cos(x²)
g = (x+1)² — chain rule: g' = 2(x+1)
Quotient rule (f'g − fg') / g²: [2x·cos(x²)·(x+1)² − sin(x²)·2(x+1)] / (x+1)⁴
Factor 2(x+1): 2[x(x+1)·cos(x²) − sin(x²)] / (x+1)³

Common Mistakes With the Quotient Rule

Mistake 1 — Reversing the Subtraction: fg'−f'g instead of f'g−fg'

❌ Wrong: (sin(x)·1 − cos(x)·x)/x² for d/dx[sin(x)/x]
✅ Correct: f'g FIRST — (cos(x)·x − sin(x)·1)/x²
The formula is (f'g − fg') / g² — NOT (fg' − f'g) / g². Reversing changes the sign of the entire result.

Mistake 2 — Writing g Instead of g² in the Denominator

❌ Wrong: (f'g−fg') / g
✅ Correct: (f'g−fg') / g² — always square the denominator. "Square the bottom."

Mistake 3 — Not Applying Chain Rule to Composite Numerator/Denominator

❌ Wrong: d/dx[sin(x²)/x] → f' = cos(x²)
✅ Correct: f = sin(x²) is composite → f' = 2x·cos(x²)

Mistake 4 — Applying Quotient Rule to Constant Denominator Unnecessarily

❌ Unnecessary: d/dx[(x²+3)/5] via quotient rule
✅ Simpler: (2x+0)/5 = 2x/5

Mistake 5 — Not Simplifying the Result

❌ Incomplete: (eˣx² − eˣ·2x)/x⁴
✅ Factor: eˣx(x−2)/x⁴ = eˣ(x−2)/x³

Limit Division Rule — Quotient Rule for Limits

The limit division rule is a limit law — completely separate from the derivative quotient rule (f'g − fg') / g².

Limit Division Rule: lim[f(x)/g(x)] = lim[f(x)] / lim[g(x)], provided lim[g(x)] ≠ 0. This divides two limiting values — it does NOT differentiate a fraction. The derivative quotient rule (f'g − fg') / g² is for differentiation only.

Example: lim[x→2] [(x²+1)/(x−3)]

lim(x²+1) / lim(x−3) = 5 / (−1) = −5
Works because lim(x−3) = −1 ≠ 0
When lim[g(x)] = 0: use L'Hôpital's rule or factor/cancel

Additional Worked Examples

d/dx[(eˣ+1) / (eˣ−1)]

f = eˣ+1, g = eˣ−1; f' = eˣ, g' = eˣ
Numerator: eˣ(eˣ−1) − (eˣ+1)eˣ = eˣ(eˣ−1−eˣ−1) = −2eˣ
Answer: −2eˣ / (eˣ−1)²

d/dx[x² / sin(x)]

f = x², g = sin(x); f' = 2x, g' = cos(x)
Quotient rule (f'g − fg') / g²: (2x·sin(x) − x²·cos(x)) / sin²(x)
Answer: x(2sin(x) − x·cos(x)) / sin²(x)

d/dx[ln(x) / eˣ]

f = ln(x), g = eˣ; f' = 1/x, g' = eˣ
Numerator: (1/x)eˣ − ln(x)eˣ = eˣ(1/x − ln(x))
Answer: (1 − x·ln(x)) / (x·eˣ)

∂/∂x[x²y / (x+y)] — Partial Quotient Rule

f = x²y (y constant), g = x+y
∂f/∂x = 2xy, ∂g/∂x = 1
Partial quotient rule: (2xy(x+y) − x²y·1) / (x+y)²
Numerator: 2x²y+2xy²−x²y = x²y+2xy² = xy(x+2y)
Answer: xy(x+2y) / (x+y)²

Frequently Asked Questions

What is the quotient rule?

The quotient rule: d/dx[f/g] = (f'g − fg') / g². Derivative of numerator times denominator minus numerator times derivative of denominator, all over denominator squared. Mnemonic: "Low d-High minus High d-Low, square the bottom and away we go." Apply the quotient rule whenever one function is divided by another.

What is the formula for the quotient rule?

The quotient rule formula: d/dx[f/g] = (f'g − fg') / g². Three equivalent forms: (f/g)' = (f'g−fg')/g² (prime), d/dx[f/g] = (f'g−fg')/g² (Leibniz), h'(x) = [f'g−fg']/g² (function notation). The order matters — f'g first, fg' subtracted. Reversing changes the sign of the entire result.

What is the "Low d-High minus High d-Low" mnemonic?

"Low d-High minus High d-Low, square the bottom and away we go" is the classic quotient rule mnemonic for (f'g − fg') / g². Low = g (denominator), High = f (numerator), d-High = f', d-Low = g'. Low·d(High) = g·f', High·d(Low) = f·g'. Subtract and divide by Low² = g² to get the quotient rule result.

What is the difference between the quotient rule and the product rule?

Product rule: d/dx[f·g] = f'g + fg' — for multiplication. Quotient rule: d/dx[f/g] = (f'g − fg') / g² — for division. The quotient rule is derived from the product rule by writing f/g = f·g⁻¹. Both give the same answer for any f/g. Choose whichever is algebraically simpler for your problem.

How do you derive tan(x) using the quotient rule?

tan(x) = sin(x)/cos(x). Apply quotient rule (f'g − fg') / g² with f=sin(x), g=cos(x), f'=cos(x), g'=−sin(x). Numerator: cos(x)·cos(x)−sin(x)·(−sin(x)) = cos²(x)+sin²(x) = 1 (Pythagorean identity). Denominator: cos²(x). Result: 1/cos²(x) = sec²(x). Therefore d/dx[tan(x)] = sec²(x).

When should you use the quotient rule vs rewriting as a product?

Both give identical results. Use the quotient rule (f'g − fg') / g² directly when the fraction form is clear. Rewrite as f·g⁻¹ and use the product rule when that simplifies the calculation. For constant denominators, simply divide: d/dx[f(x)/5] = f'(x)/5.

How do you apply the quotient rule with the chain rule?

When numerator f or denominator g is composite, apply the chain rule to find f' or g' first, then substitute into the quotient rule (f'g − fg') / g². Chain rule comes before the quotient rule assembly. Example: d/dx[sin(x²)/(x+1)²] — chain rule gives f'=2x·cos(x²) and g'=2(x+1), then apply (f'g−fg')/g².