Quotient Rule
Derivative of a quotient.
Key idea
Derivative of a quotient.
In depth
(f/g)′ = (f′g − fg′)/g². The order in the numerator matters: low d-high minus high d-low, over low squared. It follows from the product rule on f·g⁻¹.
Worked example
d/dx (x/(x + 1)) = ((1)(x + 1) − x(1))/(x + 1)² = 1/(x + 1)².
Formula
Real-world uses
- Finding how cost per unit changes as production grows
- Rates of concentration in chemistry
Common mistake
Switching the order in the numerator, giving the wrong sign.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Quotient Rule?
Derivative of a quotient. It is a topic at the advanced level in Calculus.
What is the formula for Quotient Rule?
The key formula is (f/g)′ = (f′g − fg′)/g².
Where is Quotient Rule used in real life?
Real-world examples: Finding how cost per unit changes as production grows; Rates of concentration in chemistry.
What should I learn before Quotient Rule?
Make sure you are comfortable with Product Rule, the previous topic in Calculus.
What is a common mistake with Quotient Rule?
Switching the order in the numerator, giving the wrong sign.
How can I practise Quotient Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Quotient Rule?
Move on to Chain Rule next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.