(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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