
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 vectorvalued 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 NewtonRaphson 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 














Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 