Product Rule
Derivative of product of functions
About Product Rule
The Product Rule represents derivative of product of functions. This calculus formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Calculus and Mathematical analysis. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Product Rule include Calculus problems, Physics, Engineering, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)Formula Information
Difficulty Level
Prerequisites
Discovered
17th century
Discoverer
Gottfried Leibniz
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Important Notes
Essential rule for differentiating products of functions. Remember: first times derivative of second plus second times derivative of first.
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Frequently Asked Questions
What is the product rule?
The product rule is used to find the derivative of a product of two functions. If h(x) = f(x)·g(x), then h'(x) = f'(x)·g(x) + f(x)·g'(x). In words: derivative of first times second, plus first times derivative of second. This is essential because the derivative of a product is NOT the product of derivatives.
Why can't I just multiply the derivatives?
The derivative of a product is NOT simply the product of derivatives. For example, if f(x) = x and g(x) = x, then (fg)' = (x²)' = 2x, but f'·g' = 1·1 = 1 ≠ 2x. The product rule accounts for how both functions change simultaneously, which requires the cross terms f'g + fg'.
How do I remember the product rule?
A helpful mnemonic: 'First times derivative of second, plus second times derivative of first' or 'd(uv) = u dv + v du'. Another way: (fg)' = f'g + fg'. The pattern is symmetric - both terms appear, just with derivatives swapped.
Can I use the product rule for more than two functions?
Yes! For three functions: (fgh)' = f'gh + fg'h + fgh'. For n functions, you get n terms, each with one derivative. Alternatively, you can apply the product rule repeatedly: (fgh)' = [(fg)h]' = (fg)'h + (fg)h' = (f'g + fg')h + fgh'.
When should I use the product rule vs. expanding first?
For simple products like x²·x³ = x⁵, it's easier to expand first then differentiate. But for products like x²·sin(x) or eˣ·ln(x), use the product rule. Generally, if expanding makes the expression more complicated, use the product rule.
What's the relationship between product rule and quotient rule?
The quotient rule can be derived from the product rule! Write f/g as f·(1/g), then apply product rule and chain rule. The quotient rule is: (f/g)' = (f'g - fg')/g². Notice it's similar to product rule but with a minus sign and division by g².
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Quick Details
- Category
- Calculus
- Difficulty
- Beginner
- Discovered
- 17th century
- Discoverer
- Gottfried Leibniz
- Formula ID
- product-rule