What is the derivative of csc(x)?
-csc(x)cot(x)
The derivative of csc(x) is equal to -csc(x)cot(x).
To obtain this result, we can start by using the quotient rule, which states that the derivative of a function f(x) over g(x) is given by:
(f/g)’ = (f’g – fg’)/g^2
In this case, we have:
f(x) = 1
g(x) = sin(x)
Hence, f'(x) = 0 (since the derivative of a constant is zero) and g'(x) = cos(x).
Plugging these values in the quotient rule formula, we get:
(csc(x))’ = [(0)(sin(x)) – (1)(cos(x))]/sin^2(x)
= -cos(x)/sin^2(x)
= -cot(x)csc(x)
Therefore, the derivative of csc(x) is -csc(x)cot(x).
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