## d/dx(csc(x))

### To find the derivative of csc(x) with respect to x, we can use the quotient rule and the chain rule

To find the derivative of csc(x) with respect to x, we can use the quotient rule and the chain rule.

The derivative of csc(x) can be written as:

(d/dx)(csc(x))

Let’s rewrite csc(x) in terms of sine:

csc(x) = 1/sin(x)

Now applying the quotient rule, the derivative becomes:

(d/dx)(csc(x)) = (d/dx)(1/sin(x))

Using the quotient rule, the derivative of 1/sin(x) is given by:

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

Since the derivative of a constant (1) is 0, the first term simplifies to 0.

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

Simplifying further:

[-(d/dx)(sin(x))] / (sin(x))^2

Now, applying the chain rule to the derivative of sin(x), which is cos(x), we get:

[-cos(x)] / (sin(x))^2

Finally, we can simplify this expression by using the identity cos(x) = 1/sin(x):

[-1/(sin(x))] / (sin(x))^2

Simplifying by multiplying the numerator and denominator by sin(x), we arrive at the final derivative:

-1/(sin(x))^3

So, the derivative of csc(x) with respect to x is -1/(sin(x))^3.

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