Product Rule
Derivative of a product.
Key idea
Derivative of a product.
In depth
(fg)′ = f′g + fg′. The change in a product comes from both factors changing: think of a rectangle whose width and height both grow. It is not the product of derivatives.
Worked example
d/dx (x² sin x) = 2x sin x + x² cos x.
Formula
Real-world uses
- Finding the rate of change of revenue when price and quantity both change
Common mistake
Writing (fg)′ = f′g′.
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 Product Rule?
Derivative of a product. It is a topic at the advanced level in Calculus.
What is the formula for Product Rule?
The key formula is (fg)′ = f′g + fg′.
Where is Product Rule used in real life?
Real-world examples: Finding the rate of change of revenue when price and quantity both change.
What should I learn before Product Rule?
Make sure you are comfortable with Constant and Sum Rules, the previous topic in Calculus.
What is a common mistake with Product Rule?
Writing (fg)′ = f′g′.
How can I practise Product 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 Product Rule?
Move on to Quotient 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.