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Derivatives of Exponential & Logarithmic Functions definitions

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  • Exponential Function
    An expression where a constant base is raised to a variable exponent, often modeling rapid growth or decay.
  • Natural Exponential Function
    A special exponential function with base e, notable for having a derivative equal to itself.
  • Logarithmic Function
    An expression representing the inverse of an exponential function, often used to solve for exponents.
  • Natural Logarithm
    A logarithm with base e, commonly written as ln(x), frequently used in calculus and continuous growth models.
  • Chain Rule
    A differentiation technique for composite functions, requiring multiplication by the derivative of the inner function.
  • Product Rule
    A method for differentiating the product of two functions, involving derivatives of both factors.
  • Constant Multiple Rule
    A differentiation property allowing constants to be factored out before taking the derivative.
  • Implicit Differentiation
    A process for finding derivatives when variables are mixed together, often used for equations not solved for y.
  • Base Restriction
    A requirement that the base of an exponential or logarithmic function must be positive and not equal to one.
  • Limit Definition
    The foundational approach to derivatives, involving the behavior of a function as the input approaches a specific value.
  • Power Rule
    A shortcut for differentiating expressions where a variable is raised to a constant exponent.
  • Composite Function
    A function formed by applying one function to the result of another, requiring special rules for differentiation.
  • Natural Logarithm of e
    A value always equal to one, simplifying derivatives involving base e.
  • Domain Restriction
    A limitation ensuring variables stay within valid ranges, such as x > 0 for logarithmic functions.
  • Properties of Exponents
    Rules that allow manipulation and simplification of expressions with exponents, crucial for differentiation.