d/dx csc(x)
-csc(x)cot(x)
We can start this problem by using the quotient rule of differentiation since the cosecant function could be expressed as 1/sin(x). So, applying the quotient rule,
d/dx (csc(x)) = d/dx (1/sin(x))
= -1/sin^2(x) * d/dx(sin(x))
We know that the derivative of sin(x) with respect to x is cos(x), so
d/dx (csc(x)) = -1/sin^2(x) * cos(x)
Therefore, the derivative of the cosecant function is:
d/dx (csc(x)) = -cot(x)csc(x)
where cot(x) is the cotangent function and it is equal to cos(x)/sin(x).
So, the final answer is:
d/dx (csc(x)) = -cos(x)/(sin^2(x))
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