When two variables and converge to zero at the same limit point and , they are called equivalent infinitesimal (equiv. ).
Moreover, if variables and are such that and , then:
Here is a brief proof:
Suppose there are two equivalent infinitesimals and .
For the evaluation of the indeterminate form , one can make use of the following facts about equivalent
infinitesimals (e.g., if x becomes closer to zero):[1]
For example:
In the 2nd equality, where as y become closer to 0 is used, and where is used in the 4th equality, and is used in the 5th equality.
Differentiation rules
Unless otherwise stated, all functions are functions of
real numbers (R) that return real values; although more generally, the formulae below apply wherever they are
well defined[2][3] — including the case of
complex numbers (C).[4]
Constant term rule
For any value of , where , if is the constant function given by , then .[5]
Proof
Let and . By the definition of the derivative,
This shows that the derivative of any constant function is 0.
Intuitive (geometric) explanation
The
derivative of the function at a point is the slope of the line
tangent to the curve at the point.
Slope of the constant function is zero, because the
tangent line to the constant function is horizontal and its angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
At each point, the
derivative is the slope of a
line that is
tangent to the
curve at that point. Note: the derivative at point A is
positive where green and dash–dot,
negative where red and dashed, and
zero where black and solid.
The
logarithmic derivative is another way of stating the rule for differentiating the
logarithm of a function (using the chain rule):
wherever f is positive.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.[citation needed]
Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.
The derivatives in the table above are for when the range of the inverse secant is and when the range of the inverse cosecant is
It is common to additionally define an
inverse tangent function with two arguments, Its value lies in the range and reflects the quadrant of the point For the first and fourth quadrant (i.e. ) one has Its partial derivatives are
The last series is known as
Mercator series, named after
Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia).[8] Both of these series converge for . (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.)[7]
(If n = 0, this product is an
empty product and has value 1.) It converges for for any real or complex number α.
When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1/2 and α = −1/2 give the
square root function and its
inverse:[10]
All angles are expressed in
radians. The numbers Bk appearing in the expansions of tan x are the
Bernoulli numbers. The Ek in the expansion of sec x are
Euler numbers.[12]
Hyperbolic functions
The
hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:[13]
The strict partition number sequence Q(n) has that generating function:
Integration rules
C is used for an
arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of
antiderivatives.
These formulas only state in another form the assertions in the
table of derivatives.
Integrals with a singularity
When there is a
singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the
Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in
there is a singularity at 0 and the
antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[14]
Products of functions proportional to their second derivatives
Absolute-value functions
Let f be a
continuous function, that has at most one
zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then
where sgn(x) is the
sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a
piecewise constant function):
when n is odd, and .
when for some integer n.
when for some integer n.
when for some integer n.
when for some integer n.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every
interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen
step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:
The tangent half-angle substitution relates an angle to the slope of a line.
Introducing a new variable sines and cosines can be expressed as
rational functions of and can be expressed as the product of and a rational function of as follows:
Similar expressions can be written for tan x, cot x, sec x, and csc x.
When two variables and converge to zero at the same limit point and , they are called equivalent infinitesimal (equiv. ).
Moreover, if variables and are such that and , then:
Here is a brief proof:
Suppose there are two equivalent infinitesimals and .
For the evaluation of the indeterminate form , one can make use of the following facts about equivalent
infinitesimals (e.g., if x becomes closer to zero):[1]
For example:
In the 2nd equality, where as y become closer to 0 is used, and where is used in the 4th equality, and is used in the 5th equality.
Differentiation rules
Unless otherwise stated, all functions are functions of
real numbers (R) that return real values; although more generally, the formulae below apply wherever they are
well defined[2][3] — including the case of
complex numbers (C).[4]
Constant term rule
For any value of , where , if is the constant function given by , then .[5]
Proof
Let and . By the definition of the derivative,
This shows that the derivative of any constant function is 0.
Intuitive (geometric) explanation
The
derivative of the function at a point is the slope of the line
tangent to the curve at the point.
Slope of the constant function is zero, because the
tangent line to the constant function is horizontal and its angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
At each point, the
derivative is the slope of a
line that is
tangent to the
curve at that point. Note: the derivative at point A is
positive where green and dash–dot,
negative where red and dashed, and
zero where black and solid.
The
logarithmic derivative is another way of stating the rule for differentiating the
logarithm of a function (using the chain rule):
wherever f is positive.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.[citation needed]
Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.
The derivatives in the table above are for when the range of the inverse secant is and when the range of the inverse cosecant is
It is common to additionally define an
inverse tangent function with two arguments, Its value lies in the range and reflects the quadrant of the point For the first and fourth quadrant (i.e. ) one has Its partial derivatives are
The last series is known as
Mercator series, named after
Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia).[8] Both of these series converge for . (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.)[7]
(If n = 0, this product is an
empty product and has value 1.) It converges for for any real or complex number α.
When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1/2 and α = −1/2 give the
square root function and its
inverse:[10]
All angles are expressed in
radians. The numbers Bk appearing in the expansions of tan x are the
Bernoulli numbers. The Ek in the expansion of sec x are
Euler numbers.[12]
Hyperbolic functions
The
hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:[13]
The strict partition number sequence Q(n) has that generating function:
Integration rules
C is used for an
arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of
antiderivatives.
These formulas only state in another form the assertions in the
table of derivatives.
Integrals with a singularity
When there is a
singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the
Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in
there is a singularity at 0 and the
antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[14]
Products of functions proportional to their second derivatives
Absolute-value functions
Let f be a
continuous function, that has at most one
zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then
where sgn(x) is the
sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a
piecewise constant function):
when n is odd, and .
when for some integer n.
when for some integer n.
when for some integer n.
when for some integer n.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every
interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen
step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:
The tangent half-angle substitution relates an angle to the slope of a line.
Introducing a new variable sines and cosines can be expressed as
rational functions of and can be expressed as the product of and a rational function of as follows:
Similar expressions can be written for tan x, cot x, sec x, and csc x.