The derivative of the cotangent is equal to the opposite of the square of the cosecant "-Csctwo”. This formula obeys the laws of derivative by definition and the differentiation of trigonometric functions. It is denoted as follows:
d (ctg u) = -csctwo or . du
Where "du" symbolizes the expression derived from the argument function, with respect to the independent variable.
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The procedure to develop these derivatives is quite simple. All you have to do is correctly identify the argument and the type of function it represents..
For example, the expression Ctg (f / g) has a division in its argument. This will require a differentiation regarding U / V, after developing the derivative of the cotangent.
The cotangent is the reciprocal function of the tangent. Algebraically this means that:
(1 / tg x) = ctg x
Ctg x = Cos x / Sen x
It is incorrect to say that the cotangent function is the "inverse" of the tangent. This is because the inverse tangent function by definition is arc tangent.
(Tg-1 x) = arctg x
According to Pythagorean trigonometry, the cotangent is involved in the following sections:
Ctg x = (cos x) / (sin x)
Ctgtwo x + 1 = Csctwo x
According to analytical trigonometry, it responds to the following identities:
Ctg (a + b) = (1 - tg a. Tg b) / (tg a + tg b)
Ctg (a - b) = (1 + tg a. Tg b) / (tg a - tg b)
Ctg (2a) = (1 - tgtwo a) / (2tg a)
It is necessary to analyze various characteristics of the function f (x) = ctg x in order to define the aspects necessary to study its differentiability and application.
The cotangent function is not defined on the values that make the expression "Senx" zero. Due to its equivalent Ctg x = (cos x) / (sin x), it will have an indeterminacy in all “nπ” with n belonging to the integers.
That is, in each of these values of x = nπ there will be a vertical asymptote. As you approach from the left the value of the cotangent will decrease rapidly, and as you approach from the right, the function will increase indefinitely.
The domain of the cotangent function is expressed by the set x ∈ R / x ≠ nπ, n ∈ Z. This is read as "x belonging to the set of real numbers such that x is different from nπ, with n belonging to the set of integers".
The range of the cotangent function is from minus to plus infinity. Therefore it can be concluded that its range is the set of real numbers R.
The cotangent function is periodic and its period is equal to π. In this way, the equality Ctg x = Ctg (x + nπ) is fulfilled, where n belongs to Z.
It is an odd function, since Ctg (-x) = - Ctg x. In this way it is known that the function presents a symmetry with respect to the coordinate origin. It also presents a decrease in every interval located between 2 successive vertical asymptotes.
It does not have maximum or minimum values, because its approximations to the vertical asymptotes present behaviors where the function increases or decreases indefinitely.
The zeros or roots of the cotangent function are found at odd multiples of π / 2. This means that Ctg x = 0 holds for values of the form x = nπ / 2 with n odd integer.
There are 2 ways to prove the derivative of the cotangent function.
The derivative of the cotangent function from its equivalent in sines and cosines is proved.
It is treated as the derivative of a division of functions
After deriving the factors are grouped and the aim is to emulate the Pythagorean identities
Substituting the identities and applying reciprocity, the expression is obtained
The following expression corresponds to the derivative by definition. Where the distance between 2 points of the function approaches zero.
Substituting for the cotangent we have:
Identities are applied for the sum of arguments and reciprocity
The fraction of the numerator is traditionally operated
Eliminating the opposite elements and taking a common factor, we obtain
Applying Pythagorean identities and reciprocity we have to
The elements evaluated in x are constant with respect to the limit, therefore they can leave the argument of this. Then properties of trigonometric limits are applied.
The limit is evaluated
Then it is factored until the desired value is reached
The derivative of the cotangent is thus demonstrated as the opposite of the square of the cosecant.
Based on the function f (x), define the expression f '(x)
The corresponding derivation is applied respecting the chain rule
Deriving the argument
Sometimes it is necessary to apply reciprocal or trigonometric identities to adapt the solutions.
Define the differential expression corresponding to F (x)
According to the derivation formula and respecting the chain rule
The argument is derived, while the rest remains the same
Deriving all the elements
Operating in a traditional way the products of the same base
The equal elements are added and the common factor is extracted
Signs are simplified and operated. Giving way to fully derived expression
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