Power Rule
Derivative of power function
About Power Rule
The Power Rule represents derivative of power function. 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 Power 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}[x^n] = nx^{n-1}Formula Information
Difficulty Level
Prerequisites
Discovered
17th century
Discoverer
Isaac Newton and Gottfried Leibniz
Real-World Applications
Examples
Mathematical Fields
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Important Notes
One of the most fundamental differentiation rules. Works for any real number n.
Alternative Names
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Formula Variations
Frequently Asked Questions
What is the power rule?
The power rule is the most fundamental differentiation rule: d/dx[xⁿ] = nxⁿ⁻¹. To differentiate a power function, bring down the exponent as a coefficient and reduce the exponent by 1. For example, d/dx[x³] = 3x², d/dx[x⁵] = 5x⁴, and d/dx[√x] = d/dx[x¹/²] = (1/2)x⁻¹/² = 1/(2√x).
Does the power rule work for all exponents?
Yes! The power rule works for any real number n: positive integers (x³), negative integers (x⁻²), fractions (x¹/² = √x), decimals, and even irrational numbers. The formula d/dx[xⁿ] = nxⁿ⁻¹ is universal. For example, d/dx[x^π] = πx^(π-1).
How do I use the power rule with constants?
For constants times power functions, use: d/dx[c·xⁿ] = c·nxⁿ⁻¹. The constant stays as a coefficient. For example, d/dx[5x³] = 5·3x² = 15x². For pure constants: d/dx[c] = 0 (derivative of any constant is zero).
What about negative exponents?
The power rule works for negative exponents too! d/dx[x⁻ⁿ] = -nx⁻ⁿ⁻¹. For example, d/dx[1/x] = d/dx[x⁻¹] = -1·x⁻² = -1/x². This is why 1/x differentiates to -1/x².
Can I use the power rule for polynomials?
Yes! For polynomials, apply the power rule to each term separately and add the results. For example, d/dx[3x⁴ + 2x² - 5x + 1] = 12x³ + 4x - 5. Each term uses the power rule independently, and constants differentiate to zero.
What's the relationship between power rule and integration?
The power rule for integration is the reverse: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ -1). Notice the pattern: derivative reduces power by 1, integral increases power by 1. They're inverse operations. The power rule is fundamental to both differentiation and integration.
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Quick Details
- Category
- Calculus
- Difficulty
- Beginner
- Discovered
- 17th century
- Discoverer
- Isaac Newton and Gottfried Leibniz
- Formula ID
- power-rule