Definition of Derivative (Limit Form)
The derivative of a function f(x) at a point x₀ is defined as: f'(x₀) = lim(Δ x → 0) (f(x₀ + Δ x) - f(x₀))/(Δ x) or equivalently f'(x₀) = lim(x → x₀) (f(x) - f…
Syllabus path: Functions › Preliminary Derivatives › Definition of the Derivative
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CSCA Mathematics Formula Reference
Formula and explanation
The derivative of a function f(x) at a point x₀ is defined as: f'(x₀) = lim(Δ x → 0) (f(x₀ + Δ x) - f(x₀))/(Δ x) or equivalently f'(x₀) = lim(x → x₀) (f(x) - f(x₀))/(x - x₀). Conditions: The function f(x) is defined in some neighborhood of x₀, and this limit exists. Explanation: The derivative represents the instantaneous rate of change of the function at a point, i.e., the slope of the tangent line.
Related tutorial and examples
Definition of the Derivative
Definition of the Derivative 1. Core Concepts The Derivative is the cornerstone of calculus, quantifying how fast a function changes at a specific point. Physical Meaning: Instantaneous Rate of Change (e.g., Velocity is the derivative of Displacement). Geometric Meaning: The Slop…
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