Power Rule for Derivatives: Formula, Proof and 10 Examples
The power rule says d/dx xⁿ = n·xⁿ⁻¹. Learn when it applies, how to use it with roots, fractions and negative exponents, with 10 examples.
Guides & worked examples
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Read article → Power Rule for Derivatives: Formula, Proof and 10 Examples
Derivative of cos²(x): −sin(2x) with Two Methods
The derivative of cos²(x) is −2sin(x)cos(x) = −sin(2x). Two methods, plus sin²(x), cos(x²) and cos²(3x) worked examples.
Derivative of x^x: Logarithmic Differentiation Explained
The derivative of x^x is x^x(ln x + 1). Step-by-step logarithmic differentiation, why the power rule fails, and more examples.
Derivative of e^(2x): 2e^(2x) Step by Step
The derivative of e^(2x) is 2e^(2x). Learn why with the chain rule, plus e^(kx), e^(x²), e^(−x) and x·e^(2x) examples.
Derivative of √x: 1/(2√x) with Proof and Examples
The derivative of √x is 1/(2√x). See the power rule method, the limit proof, chain rule examples like √(x²+1), and common errors.
Derivative of arcsin(x) = 1/√(1−x²): Proof and Examples
The derivative of arcsin(x) is 1/√(1−x²) for −1 < x < 1. Proof, chain rule examples, and the arccos version with its minus sign.
Derivative of arctan(x) = 1/(1+x²): Proof and Examples
The derivative of arctan(x) is 1/(1+x²). Step-by-step proof with implicit differentiation, chain rule examples and the link to integrals.
Derivative of sec(x) = sec(x)tan(x): Proof and Examples
The derivative of sec(x) is sec(x)tan(x). See the proof from 1/cos(x), the chain rule form, second derivative and worked examples.
Derivative of tan(x) = sec²(x): Proof and Examples
The derivative of tan(x) is sec²(x). Learn the quotient rule proof, the chain rule version for tan(g(x)), and worked examples.
Derivative of ln(x): Formula, Proof and Examples
The derivative of ln(x) is 1/x. See two short proofs, the chain rule version for ln(g(x)), worked examples and common mistakes.