## Derivative of csc x

### The derivative of the cosecant (csc) function, denoted as d/dx(csc x), can be found using the quotient rule

The derivative of the cosecant (csc) function, denoted as d/dx(csc x), can be found using the quotient rule.

The cosecant function, csc(x), is defined as the reciprocal of the sine function:

csc(x) = 1 / sin(x)

Using the quotient rule, the derivative of csc(x) can be found as follows:

d/dx(csc x) = [d/dx(1)]*sin(x) – [1]*[d/dx(sin(x))] / (sin(x))^2

To find the derivative of the sine function, we can use the chain rule, which states that the derivative of sin(u) with respect to x is equal to the derivative of sin(u) with respect to u times the derivative of u with respect to x:

d/dx(sin(x)) = cos(x)

Substituting this derivative back into the quotient rule equation, we have:

d/dx(csc x) = [0]*sin(x) – [1]*cos(x) / (sin(x))^2

Simplifying further, we get:

d/dx(csc x) = -cos(x) / (sin(x))^2

Therefore, the derivative of the cosecant function, d/dx(csc x), is equal to -cos(x) / (sin(x))^2.

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