Differentiation of cosec x

Integral of Cosec x To find the integral of cosec x, we will have to use trigonometry. Cosec x is the reciprocal of sin x and cot x is equal to (sin x)/(cos x). We can do the integration of cosecant x in multiple methods such as: • By substitution method • By partial fractions • By trigonometric formulas We have different formulas for integration of cosec x and let us derive each of them using each of the above-mentioned methods. Also, we will solve some examples related to the integral of cosec x. 1. 2. 3. 4. 5. What is the Integral of Cosec x? The ∫ cosec x dx (or) ∫ csc x dx and its value is ln |cosec x - cot x| + C. This is also known as the antiderivative of ∫ cosec x dx = ln |cosec x - cot x| + C. Here "ln" represents the natural • ∫ cosec x dx = ln |cosec x - cot x| + C [OR] • ∫ cosec x dx = - ln |cosec x + cot x| + C [OR] • ∫ cosec x dx = (1/2) ln | (cos x - 1) / (cos x + 1) | + C [OR] • ∫ cosec x dx = ln | tan (x/2) | + C We use one of these formulas according to necessity. We will prove each of these formulas in each of the mentioned methods. Integral of Cosec x by Substitution Method We can find the integral of cosec x proof by ∫ cosec x dx = ∫ cosec x · (cosec x - cot x) / (cosec x - cot x) dx = ∫ (cosec 2x - cosec x cot x) / (cosec x - cot x) dx Now assume that cosec x - cot x = u. Then (-cosec x cot x + cosec 2x) dx = du. Substituting these values in the above integral, ∫ cosec x dx = ∫ du / u = ln |u| + C Substituting u = cosec x - cot x back here, ∫ cosec x...

• • • • • The Differentiation of function in Limit form As per $\dfrac$ Therefore, it is proved that the derivative of cosecant function is equal to the negative product of cosecant and cotangent functions.

Derivatives of the Trigonometric Functions Formulas of the Formulae For The Derivatives of Trigonometric Functions 1 - Derivative of sin x The derivative of f(x) = sin x is given by f '(x) = cos x 2 - Derivative of cos x The derivative of f(x) = cos x is given by f '(x) = - sin x 3 - Derivative of tan x The derivative of f(x) = tan x is given by f '(x) = sec 2 x 4 - Derivative of cot x The derivative of f(x) = cot x is given by f '(x) = - csc 2 x 5 - Derivative of sec x The derivative of f(x) = sec x tan x is given by f '(x) = sec x tan x 6 - Derivative of csc x The derivative of f(x) = csc xis given by f '(x) = - csc x cot x Examples Using the Derivatives of Trigonometric Functions Example 1 Find the first derivative of f(x) = x sin x Solution to Example 1: • Let g(x) = x and h(x) = sin x, function f may be considered as the product of functions g and h: f(x) = g(x) h(x). Hence we use the product rule, f '(x) = g(x) h '(x) + h(x) g '(x), to differentiate function f as follows f '(x) = x cos x + sin x * 1 = x cos x + sin x Example 2 Find the first derivative of f(x) = tan x + sec x Solution to Example 2: • Let g(x) = tan x and h(x) = sec x, function f may be considered as the sum of functions g and h: f(x) = g(x) + h(x). Hence we use the sum rule, f '(x) = g '(x) + h '(x), to differentiate function f as follows f '(x) = sec 2 x + sec x tan x = sec x (sec x + tan x) Example 3 Find the first derivative of f(x) = sin x / [ 1 + cos x ] Solution to Example 3: • Let g(x) = sin x...

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2. Derivatives of Csc, Sec and Cot Functions by M. Bourne By using the quotient rule and trigonometric identities, we can obtain the following derivatives: `(d(csc x))/(dx)=-csc x cot x` `(d(sec x))/(dx)=sec x tan x` `(d(cot x))/(dx)=-csc^2 x` In words, we would say: The derivative of `csc x` is `-csc x cot x`, The derivative of `sec x` is `sec x tan x` and The derivative of `cot x` is `-csc^2 x`. Explore animations of these functions with their derivatives here: If u = f( x) is a function of x, then by using the chain rule, we have: `(d(csc u))/(dx)=-csc u\ cot u(du)/(dx)` `(d(sec u))/(dx)=sec u\ tan u(du)/(dx)` `(d(cot u))/(dx)=-csc^2u(du)/(dx)` Example 1 Find the derivative of s = sec(3 t + 2). Answer This is an implicit function. 3 cot( x + y) = cos y 2 For the left hand side, we put u = x + y. Differentiating 3 cot u gives us: `3(-csc^2 u)((du)/(dx))` Substituting for `u` and performing the `(du)/(dx)` part gives us: `-3 csc^2(x+y)(1+(dy)/(dx))` On the right hand side, we let u = y 2. Differentiating `cos u` gives us: `(-sin u)((du)/(dx))` Substituting for `u` and performing the `(du)/(dx)` part gives us: `(-sin y^2)(2y(dy)/(dx))` We put both sides together: `-3 csc^2(x+y)(1+(dy)/(dx))` `=(-sin y^2)(2y(dy)/(dx))` Expanding gives: `-3 csc^2(x+y)` `-3 csc^2(x+y)(dy)/(dx)` `=-2y sin y^2(dy)/(dx)` Adding `2y sin y^2(dy)/(dx)` to both sides: `-3 csc^2(x+y)` `-3 csc^2(x+y)(dy)/(dx)` `+2y sin y^2(dy)/(dx)` `=0` Adding `3 csc^2(x+y)` to both sides: `-3 csc^2(x+y)(dy)/(dx)` `+...

Introduction to the derivative of cscx Derivatives have a wide range of applications in almost every field of engineering and science. The cscx derivative can be calculated by following the rules of differentiation. Or, we can directly find the derivative of cosecant x by applying the first principle of differentiation. In this article, you will learn what the What is the derivative of cscx? The derivative of cscx or cosecant x is a commonly searched topic in calculus. The derivative of cscx with respect to x can be found using the formula d/dx(csc x) = -csc x.cot x, which involves taking the derivative of both the sine and cosine functions. This formula can be useful for finding the rate of change of csc x, which is the reciprocal of the sine function, in various applications. In a right triangle, the cosecant x is equal to the ratio of the hypotenuse to the opposite side, or 1/sin x. By using this relationship, we can find the value of cosecant x for any given angle. Derivative of csc(x) formula The derivative of cscx, also known as cosecant x, can be found using the formula: d/dx (csc x) = -cot x . csc x. This formula can be useful in finding the rate of change of the cosecant function, which is the reciprocal of the sine function. Remember that the derivative of -cscx is the same as the derivative of cscx. Using this formula, you can easily find the derivative of csc x or solve problems like finding the second derivative f(x) = csc(x). How to calculate csc derivative? ...

Introduction to the derivative of cscx Derivatives have a wide range of applications in almost every field of engineering and science. The cscx derivative can be calculated by following the rules of differentiation. Or, we can directly find the derivative of cosecant x by applying the first principle of differentiation. In this article, you will learn what the What is the derivative of cscx? The derivative of cscx or cosecant x is a commonly searched topic in calculus. The derivative of cscx with respect to x can be found using the formula d/dx(csc x) = -csc x.cot x, which involves taking the derivative of both the sine and cosine functions. This formula can be useful for finding the rate of change of csc x, which is the reciprocal of the sine function, in various applications. In a right triangle, the cosecant x is equal to the ratio of the hypotenuse to the opposite side, or 1/sin x. By using this relationship, we can find the value of cosecant x for any given angle. Derivative of csc(x) formula The derivative of cscx, also known as cosecant x, can be found using the formula: d/dx (csc x) = -cot x . csc x. This formula can be useful in finding the rate of change of the cosecant function, which is the reciprocal of the sine function. Remember that the derivative of -cscx is the same as the derivative of cscx. Using this formula, you can easily find the derivative of csc x or solve problems like finding the second derivative f(x) = csc(x). How to calculate csc derivative? ...

Derivatives of the Trigonometric Functions Formulas of the Formulae For The Derivatives of Trigonometric Functions 1 - Derivative of sin x The derivative of f(x) = sin x is given by f '(x) = cos x 2 - Derivative of cos x The derivative of f(x) = cos x is given by f '(x) = - sin x 3 - Derivative of tan x The derivative of f(x) = tan x is given by f '(x) = sec 2 x 4 - Derivative of cot x The derivative of f(x) = cot x is given by f '(x) = - csc 2 x 5 - Derivative of sec x The derivative of f(x) = sec x tan x is given by f '(x) = sec x tan x 6 - Derivative of csc x The derivative of f(x) = csc xis given by f '(x) = - csc x cot x Examples Using the Derivatives of Trigonometric Functions Example 1 Find the first derivative of f(x) = x sin x Solution to Example 1: • Let g(x) = x and h(x) = sin x, function f may be considered as the product of functions g and h: f(x) = g(x) h(x). Hence we use the product rule, f '(x) = g(x) h '(x) + h(x) g '(x), to differentiate function f as follows f '(x) = x cos x + sin x * 1 = x cos x + sin x Example 2 Find the first derivative of f(x) = tan x + sec x Solution to Example 2: • Let g(x) = tan x and h(x) = sec x, function f may be considered as the sum of functions g and h: f(x) = g(x) + h(x). Hence we use the sum rule, f '(x) = g '(x) + h '(x), to differentiate function f as follows f '(x) = sec 2 x + sec x tan x = sec x (sec x + tan x) Example 3 Find the first derivative of f(x) = sin x / [ 1 + cos x ] Solution to Example 3: • Let g(x) = sin x...

• • • • • The Differentiation of function in Limit form As per $\dfrac$ Therefore, it is proved that the derivative of cosecant function is equal to the negative product of cosecant and cotangent functions.

Integral of Cosec x To find the integral of cosec x, we will have to use trigonometry. Cosec x is the reciprocal of sin x and cot x is equal to (sin x)/(cos x). We can do the integration of cosecant x in multiple methods such as: • By substitution method • By partial fractions • By trigonometric formulas We have different formulas for integration of cosec x and let us derive each of them using each of the above-mentioned methods. Also, we will solve some examples related to the integral of cosec x. 1. 2. 3. 4. 5. What is the Integral of Cosec x? The ∫ cosec x dx (or) ∫ csc x dx and its value is ln |cosec x - cot x| + C. This is also known as the antiderivative of ∫ cosec x dx = ln |cosec x - cot x| + C. Here "ln" represents the natural • ∫ cosec x dx = ln |cosec x - cot x| + C [OR] • ∫ cosec x dx = - ln |cosec x + cot x| + C [OR] • ∫ cosec x dx = (1/2) ln | (cos x - 1) / (cos x + 1) | + C [OR] • ∫ cosec x dx = ln | tan (x/2) | + C We use one of these formulas according to necessity. We will prove each of these formulas in each of the mentioned methods. Integral of Cosec x by Substitution Method We can find the integral of cosec x proof by ∫ cosec x dx = ∫ cosec x · (cosec x - cot x) / (cosec x - cot x) dx = ∫ (cosec 2x - cosec x cot x) / (cosec x - cot x) dx Now assume that cosec x - cot x = u. Then (-cosec x cot x + cosec 2x) dx = du. Substituting these values in the above integral, ∫ cosec x dx = ∫ du / u = ln |u| + C Substituting u = cosec x - cot x back here, ∫ cosec x...