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