Derivative of csc(x) using the quotient rule | -1 / sin(x)sec(x)

(d/dx) csc(x)

To find the derivative of csc(x), we can use the quotient rule

To find the derivative of csc(x), we can use the quotient rule. The quotient rule states that for functions u(x) and v(x), the derivative of u(x)/v(x) is given by [v(x) * u'(x) – u(x) * v'(x)] / [v(x)]^2.

In this case, u(x) = 1 and v(x) = sin(x). Therefore, u'(x) = 0 (since the derivative of a constant is zero) and v'(x) = cos(x) (since the derivative of sin(x) is cos(x)).

Now we can apply the quotient rule:

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

So, the derivative of csc(x) is -1 / sin(x)sec(x).

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