(d/dx) cot(x)
-csc^2(x)
Let’s start by recalling the definition of the cotangent function:
cot(x) = cos(x) / sin(x)
Now we can use the quotient rule to find the derivative:
(d/dx) cot(x)
= (d/dx) cos(x)/sin(x)
= [(sin(x) d/dx cos(x)) – (cos(x) d/dx sin(x))] / sin^2(x)
= [(-sin(x))(-sin(x)) – (cos(x))(cos(x))]/sin^2(x) [applying d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x)]
= (-sin^2(x)-cos^2(x))/sin^2(x)
= -1/(sin(x)cos(x))
Therefore,
(d/dx) cot(x) = -1/(sin(x)cos(x))
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