What is the derivative of csc(x) with respect to x?

The derivative of csc(x), which is the cosecant function, involves using the properties of trigonometric identities and derivatives. The cosecant function can be expressed as the reciprocal of the sine function, so csc(x) = 1/sin(x).

To find the derivative of csc(x) with respect to x, we apply the quotient rule or the chain rule. Using the chain rule, we have:

1. The derivative of sin(x) is cos(x).

2. Consequently, applying the derivative of the reciprocal function, we can derive the formula for csc(x):

• If y = 1/sin(x), then dy/dx = -1/sin^2(x) * cos(x) using the chain rule.

Rearranging this, we can express it in terms of csc(x) and cot(x). Since cot(x) is defined as cos(x)/sin(x), we arrive at the derivative being expressed as:

dy/dx = -csc(x) * cot(x).

This aligns perfectly with the first choice, confirming that the correct derivative of csc(x) with respect to x is indeed -csc(x)cot(x). This understanding of the relationship between the derivative of trigon