Derivative of Cosecant x

We now understand the derivative of csc x. It is commonly represented as d/dx (csc x) or (csc x)', and its value is -csc(x)cot(x). The function csc x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Cosecant Function: (csc(x) = 1/sin(x)).

Quotient Rule: Rule for differentiating csc(x) (since it consists of 1/sin(x)).

Cotangent Function: cot(x) = cos(x)/sin(x).

The derivative of csc x can be denoted as d/dx (csc x) or (csc x)'. The formula we use to differentiate csc x is: d/dx (csc x) = -csc(x)cot(x) (or) (csc x)' = -csc(x)cot(x)

The formula applies to all x where sin(x) ≠ 0

We can derive the derivative of csc x using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation.

There are several methods we use to prove this, such as:

• By First Principle
   
• Using Chain Rule
   
• Using Product Rule

We will now demonstrate that the differentiation of csc x results in -csc(x)cot(x) using the above-mentioned methods:

By First Principle

The derivative of csc x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of csc x using the first principle, we will consider f(x) = csc x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = csc x, we write f(x + h) = csc (x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [csc(x + h) - csc x] / h = limₕ→₀ [ [1/sin (x + h)] - [1/sin x] ] / h = limₕ→₀ [ [sin x - sin (x + h)] / [sin(x + h) · sin x] ] / h We now use the formula sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2). f'(x) = limₕ→₀ [ -2 cos((2x + h)/2) sin(h/2) ] / [ h sin(x + h) · sin x] = limₕ→₀ [ -2 cos(x + h/2) sin(h/2) / h ] · [ 1 / sin(x + h) · sin x ] Using limit formulas, limₕ→₀ (sin(h/2)/(h/2)) = 1. f'(x) = -2 cos(x) / [sin²x] As cosecant is the reciprocal of sine, we have, f'(x) = -csc(x)cot(x). Hence, proved.

Using Chain Rule

To prove the differentiation of csc x using the chain rule, We use the formula: Csc x = 1/sin x Consider f(x) = 1 and g(x)= sin x So we get, csc x = f(x)/g(x) By quotient rule: d/dx [f(x) / g(x)] = [f '(x) g(x) - f(x) g'(x)] / [g(x)]²… (1) Let’s substitute f(x) = 1 and g(x) = sin x in equation (1), d/dx (csc x) = [(0) (sin x) - (1) (cos x)] / (sin x)² = -cos x / sin² x Here, we use the formula: cos x / sin x = cot x (cotangent identity) Substituting this into the equation, d/dx (csc x) = -csc(x)cot(x)

Using Product Rule

We will now prove the derivative of csc x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, Csc x = 1/sin x csc x = (sin x)⁻¹ Given that, u = (sin x)⁻¹ Using the product rule formula: d/dx [u] = -1(sin x)⁻² (cos x) = -cos x / sin² x Here we use the chain rule: v = cos x / sin x = cot x Substituting u = (sin x)⁻¹ and using the product rule, we find: d/dx (csc x) = -csc(x)cot(x).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like csc(x).

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of csc(x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

When x is 0, the derivative is undefined because csc(x) has a vertical asymptote there.

When x is π/2, the derivative of csc x = -csc(π/2)cot(π/2), which is 0.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Cosecant Function: A trigonometric function that is the reciprocal of the sine function. It is typically represented as csc x.

• Cotangent Function: One of the primary six trigonometric functions and is written as cot x.

• First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.

• Asymptote: The function goes near a line without intersecting or crossing it. This line is known as an asymptote.