## derv csc (x)

### The derivative of csc(x), which is the reciprocal of the sine function, can be found using the quotient rule

The derivative of csc(x), which is the reciprocal of the sine function, can be found using the quotient rule. The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is given by (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2.

In the case of csc(x), we can rewrite it as 1/sin(x). Applying the quotient rule, we have:

u(x) = 1

v(x) = sin(x)

Now, let’s find the derivatives of u(x) and v(x):

u'(x) = 0 (derivative of a constant is zero)

v'(x) = cos(x) (derivative of sin(x) is cos(x))

Plugging these values into the quotient rule formula:

csc'(x) = (cos(x) * 1 – 1 * sin(x)) / (sin(x))^2

Simplifying further:

csc'(x) = cos(x) – sin(x) / sin^2(x)

Now, if you want to simplify this further, you can use the trigonometric identity sin^2(x) + cos^2(x) = 1. Rearranging this identity, we get sin^2(x) = 1 – cos^2(x).

Plugging this into the derivative formula:

csc'(x) = cos(x) – sin(x) / (1 – cos^2(x))

And there you have the derivative of csc(x) in simplified form.

Note: It’s important to keep in mind the domain of the function. The derivative of csc(x) is undefined at values where sin(x) = 0, which occurs at x = nπ, where n is an integer.

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