Derivative of csc²(x)

We now understand the derivative of csc²(x). It is commonly represented as d/dx (csc²(x)) or (csc²(x))', and its value is -2csc²(x)cot(x). The function csc²(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 csc²(x) (since it is a power of a trigonometric function).

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

The derivative of csc²(x) can be denoted as d/dx (csc²(x)) or (csc²(x))'. The formula we use to differentiate csc²(x) is: d/dx (csc²(x)) = -2csc²(x)cot(x) (or) (csc²(x))' = -2csc²(x)cot(x)

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

We can derive the derivative of csc²(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 csc²(x) results in -2csc²(x)cot(x) using the above-mentioned methods:

By First Principle

The derivative of csc²(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of csc²(x) using the first principle, we will consider f(x) = csc²(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) = csc²(x), we write f(x + h) = csc²(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [csc²(x + h) - csc²(x)] / h = limₕ→₀ [1/sin²(x + h) - 1/sin²(x)] / h = limₕ→₀ [[sin²(x) - sin²(x + h)] / [sin²(x)sin²(x + h)]] / h We now use the formula sin²(A) - sin²(B) = (A - B)(A + B). f'(x) = limₕ→₀ [(sin(x + h) - sin(x))(sin(x + h) + sin(x))] / [h sin²(x)sin²(x + h)] = limₕ→₀ [(sin h) cos((x + x + h)/2) 2cos((x - x - h)/2)] / [h sin²(x)sin²(x + h)] = limₕ→₀ (sin h)/ h · limₕ→₀ [cos((x + x + h)/2) · 2cos((x - x - h)/2)] / [sin²(x)sin²(x + h)] Using limit formulas, limₕ→₀ (sin h)/ h = 1. f'(x) = -2csc²(x)cot(x) Hence, proved.

Using Chain Rule

To prove the differentiation of csc²(x) using the chain rule, We use the formula: csc²(x) = (csc(x))² Let u = csc(x), so we have u² By chain rule: d/dx (u²) = 2u u' Let’s substitute u = csc(x) and u' = -csc(x)cot(x), d/dx (csc²(x)) = 2(csc(x))(-csc(x)cot(x)) = -2csc²(x)cot(x)

Using Product Rule

We will now prove the derivative of csc²(x) using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, csc²(x) = (csc(x))(csc(x)) Given that, u = csc(x) and v = csc(x) Using the product rule formula: d/dx [u.v] = u'.v + u.v' u' = d/dx (csc(x)) = -csc(x)cot(x) (substitute u = csc(x)) v' = d/dx (csc(x)) = -csc(x)cot(x) (substitute v = csc(x))

Again, use the product rule formula: d/dx (csc²(x)) = u'.v + u.v' Let’s substitute u = csc(x), u' = -csc(x)cot(x), v = csc(x), and v' = -csc(x)cot(x) When we simplify each term: We get, d/dx (csc²(x)) = -csc(x)cot(x)csc(x) + csc(x)(-csc(x)cot(x)) = -2csc²(x)cot(x) Thus: d/dx (csc²(x)) = -2csc²(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 csc²(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 csc²(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 nπ (where n is any integer), the derivative is undefined because csc(x) has a vertical asymptote there.

When x is π/4, the derivative of csc²(x) = -2csc²(π/4)cot(π/4), which is -2.

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

• Cosecant Function: The cosecant function is one of the primary six trigonometric functions and is written as csc(x).

• Cotangent Function: A trigonometric function that is the reciprocal of the tangent function. It is typically represented 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.