(d/dx) csc(x)
-csc(x)cot(x)
We can use the quotient rule and chain rule to find the derivative of csc(x) with respect to x:
(csc(x))’ = (-cot(x)csc(x))’.
Using the quotient rule on (-cot(x)csc(x)), we get:
(-cot(x)csc(x))’ = (-cot(x))’csc(x) + cot(x)(csc(x))’
Since (-cot(x))’ = csc^2(x) by the chain rule and (csc(x))’ = -csc(x)cot(x) by the derivative of sine, we can substitute these values into the equation:
(csc(x))’ = (-cot(x))’csc(x) + cot(x)(csc(x))’
(csc(x))’ = csc^2(x)*csc(x) + cot(x)*(-csc(x)*cot(x))
(csc(x))’ = csc^3(x) – cot^2(x)csc(x)
Therefore, the derivative of csc(x) with respect to x is csc^3(x) – cot^2(x)csc(x).
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