The Quotient Rule
Learn to differentiate a quotient of two functions and simplify the results effectively.
What Is the Quotient Rule?
The quotient rule is used to differentiate a function that is the ratio (quotient) of two other functions. If y = u/v, where both u and v are functions of x, and v is not zero:
If y = u/v, then:
dy/dx = (u'v - uv') / v2
In words: "the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared." Note the subtraction -- this is the key difference from the product rule.
Step-by-Step Method
Identify u (numerator function) and v (denominator function).
Find u' and v' (differentiate numerator and denominator separately).
Substitute into the formula: dy/dx = (u'v - uv') / v2.
Expand the numerator and simplify. Factor where possible.
Important: The order matters in the quotient rule! u'v - uv' is NOT the same as uv' - u'v. A common mnemonic is "low d-high minus high d-low, over low squared", where "high" is the numerator and "low" is the denominator.
When to Use vs Rewrite
Sometimes you can avoid the quotient rule by rewriting the expression using negative indices. For example:
Use Quotient Rule
When both numerator and denominator are more complex functions, e.g. y = sin(x) / (x2 + 1)
Rewrite Instead
When the denominator is simple, e.g. y = x3/x2 = x (just simplify first)
Key Vocabulary
Quotient
The result of dividing one quantity by another. In y = u/v, y is the quotient of u and v.
Numerator
The top part of a fraction. In the quotient rule, this is the function u.
Denominator
The bottom part of a fraction. In the quotient rule, this is the function v, which must not equal zero.
Simplify
To reduce an expression to its simplest form by expanding, cancelling, and factorising.
Worked Examples
Differentiate y = x2 / (x + 1)
Step 1: u = x2, v = x + 1.
Step 2: u' = 2x, v' = 1.
Step 3: dy/dx = (2x(x + 1) - x2(1)) / (x + 1)2
Step 4: = (2x2 + 2x - x2) / (x + 1)2 = (x2 + 2x) / (x + 1)2
Answer: dy/dx = x(x + 2) / (x + 1)2.
Differentiate y = sin(x) / x
Step 1: u = sin(x), v = x.
Step 2: u' = cos(x), v' = 1.
Step 3: dy/dx = (cos(x) × x - sin(x) × 1) / x2
Answer: dy/dx = (x cos(x) - sin(x)) / x2.
Differentiate y = ex / (2x + 3)
Step 1: u = ex, v = 2x + 3.
Step 2: u' = ex, v' = 2.
Step 3: dy/dx = (ex(2x + 3) - ex × 2) / (2x + 3)2
Step 4: = ex(2x + 3 - 2) / (2x + 3)2 = ex(2x + 1) / (2x + 3)2
Answer: dy/dx = ex(2x + 1) / (2x + 3)2.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
In the quotient rule dy/dx = (u'v - uv') / v2, the denominator of the result is always:
Question 2
Differentiate y = (3x + 1) / x2. What is dy/dx?
Question 3
Differentiate y = x / (x - 1). What is dy/dx?
Question 4
What is the key difference between the product rule and the quotient rule formulae?
Question 5
Differentiate y = ex / x. What is dy/dx?
Key Concepts Summary
- ● The quotient rule is dy/dx = (u'v - uv') / v2.
- ● The minus sign and order are critical -- do not confuse with the product rule.
- ● Always simplify the numerator and factorise where possible.
- ● Consider whether you can rewrite using negative indices and use the chain rule instead.
- ● The denominator of the result is always v2 (the bottom squared).