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Derivatives as Functions definitions

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  • Derivative
    Represents the slope of a tangent line to a function at any given point, found using a specific limit process.
  • Tangent Line
    A straight line that touches a curve at a single point and has the same slope as the curve at that point.
  • Slope
    A measure of steepness, indicating how much a function rises or falls as its input changes.
  • Limit
    A mathematical approach used to find the value a function approaches as the input gets infinitely close to a specific point.
  • Prime Notation
    A shorthand symbol, often an apostrophe, used to denote the derivative of a function, such as f'(x).
  • General Equation
    An expression that gives the derivative for all possible input values, allowing calculation at any point.
  • Definition of Derivative
    A formula involving a limit that calculates the instantaneous rate of change for a function.
  • Function
    A rule that assigns each input exactly one output, often written as f(x).
  • Variable
    A symbol, such as x or h, representing a quantity that can change within a mathematical expression.
  • Expression
    A combination of numbers, variables, and operations that represents a mathematical relationship.
  • Simplification
    The process of rewriting a mathematical expression in a more concise or manageable form.
  • Expansion
    The process of multiplying out expressions, such as turning (x + h)² into x² + 2xh + h².
  • Cancellation
    The process of removing common factors from the numerator and denominator to simplify a fraction.
  • Instantaneous Rate of Change
    The value describing how a function changes at a single point, found using the derivative.
  • Input Value
    A specific number substituted for the variable in a function to evaluate its output or slope.