DERIVATIVE OF COTANGENT: CALCULATION, PROOF, EXERCISES - MATHS - 2025

The derivative of the cotangent is equal to the opposite of the square of the cosecant "-Csc ² ". This formula obeys the laws of derivative by definition and the differentiation of trigonometric functions. It is denoted as follows:

d (ctg u) = -csc ² u. du

Where "du" symbolizes the expression derived from the argument function, with respect to the independent variable.

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How is it calculated?

The procedure to develop these derivatives is quite simple. It is enough just to 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 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)

Ctg ² x + 1 = Csc ² 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 - tg ² a) / (2tg a)

Characteristics of the cotangent function

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.

Vertical asymptotes

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.

Domain

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".

Rank

The range of the cotangent function is from minus to plus infinity. Therefore, it can be concluded that its rank is the set of real numbers R.

Frequency

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.

Behavior

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.

Demonstration

There are 2 ways to prove the derivative of the cotangent function.

Trigonometric differential proof

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

Proof by definition of derivative

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.

Solved exercises

Exercise 1

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.

Exercise 2

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 the fully derived expression

References

1. Trigonometric Series, Volume 1. A. Zygmund. Cambridge University Press, 2002
2. Calculus of a Single Variable. Ron Larson, Bruce H. Edwards. Cengage Learning, Nov 10 2008
3. Calculus with trigonometry and analytic geometry. John H. Saxon, John Saxon, Frank Wang, Diana Harvey. Saxon Publishers, 1988
4. Multivariable Analysis. Satish Shirali, Harkrishan Lal Vasudeva. Springer Science & Business Media, Dec 13. 2010
5. System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg. John Wiley & Sons, Mar 7 2012
6. Calculus: Mathematics and Modeling. William Bauldry, Joseph R. Fiedler, Frank R. Giordano, Ed Lodi, Rick Vitray. Addison Wesley Longman, Jan 1 1999