Functions Contents - F
Differential calculus
       Differential Calculus - Derivatives and differentials
      The derivative of a function
         Definition of the derivative of a function
      Tangent to a curve
      The equation of the line tangent to the given curve at the given point
         Determining the derivative of a function as the limit of the difference quotient
         The equation of the line tangent to a curve at the given point example
      Determining the lines tangent to the graph of a function from a point outside the function
         Determining the lines tangent to the graph of a function from a point outside the function example
      The derivative as a function
         The derivative function
         Differentiation, determining (or deriving) derivative of a function
      Derivatives of basic or elementary functions
         Determining the derivative of a function as the limit of the difference quotient
         The derivative of the power function
         The derivative of the linear function
         The derivative of a constant
         The derivative of the sine function
         The derivative of the cosine function
         The derivative of the exponential function
         The derivative of the logarithmic function
      Differentiation rules
         The derivative of the sum or difference of two functions
         The product rule
         A constant times a function rule
         Derivatives of functions examples
         The quotient rule
         The derivative of the tangent function, use of the quotient rule
         The derivative of the cotangent function, use of the quotient rule
         The chain rule
         Differentiation using the chain rule examples
      Table of derivatives of elementary functions
         Differentiation rules
         Table of derivatives
      The chain rule applications
      Implicit differentiation
         Implicit differentiation examples
      Generalized power rule
         Generalized power rule examples
      Logarithmic differentiation
         Logarithmic differentiation examples
      Derivative of a composite exponential function
         Use of the logarithmic differentiation
         Derivatives of composite functions examples
      Derivatives of the hyperbolic functions
         Derivatives of inverse hyperbolic functions
      Derivative of the inverse function
         Derivatives of the inverse trigonometric functions
      Derivative of parametric functions, parametric derivatives
         Derivative of parametric functions example
      Derivative of vector-valued functions
      Applications of the derivative
      Tangent, normal, subtangent and subnormal
         Property of the parabola
         Property of power functions
         Property of the exponential function
      Angle between two curves
         Angle between two curves examples
      Differential of a function
         Use of differential to approximate the value of a function
         Rules for differentials
         Differentials of some basic functions
      Higher order derivatives and higher order differentials
      Higher order derivatives
         Higher order derivatives examples
         Higher derivative formula for the product - Leibniz formula
         Higher derivatives of composite functions
         Higher derivatives of composite functions examples
         Higher derivatives of implicit functions
         Higher derivatives of implicit functions examples
         Higher derivatives of parametric functions
         Higher derivatives of parametric functions examples
      Higher order differentials
         Higher order differentials examples
      Applications of differentiation - the graph of a function and its derivative
      Definition of increasing and decreasing
         Increasing/decreasing test
      Rolle's theorem
      The mean value theorem
         Generalization of the mean value theorem
         Cauchy's mean value theorem or generalized mean value theorem
      L'Hospital's rule - limits of indeterminate forms
         Applications of L'Hospital's rule - evaluation of limits of indeterminate forms, examples
      Generalization of the mean value theorem, concavity of the graph of a function
      Concavity of the graph of a function
         Concave up and concave down definition
         Points of inflection
         Points of inflection and concavity of the sine function
         Points of inflection and concavity of the cubic polynomial
      Approximate solution to an equation, Newton's method (or the Newton-Raphson method)
         Use of Newton's method, example
      Taylor's theorem (Taylor's formula) - The extended mean value theorem
         The proof of Thaylor's theorem
         Maclaurin's formula or Maclaurin's theorem
      Representing polynomial using Maclaurin's and Taylor's formula
         Representing polynomial using Maclaurin's and Taylor's formula examples
      The approximation of the exponential function by polynomial using Taylor's or Maclaurin's formula
         Properties of the power series expansion of the exponential function
      The approximation of the sine function by polynomial using Taylor's or Maclaurin's formula
         Properties of the power series expansion of the sine function
      Extreme points, local (or relative) maximum and local minimum
         The first derivative test
         The second derivative test and concavity
      Finding and classifying critical or stationary points
         Finding extreme points
         Points of inflection
         Finding points of inflection
         Finding and classifying critical or stationary points examples
   
 
 
 
   
   
 
   
 
 
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