d/dx cotx
derivative of cot(x) is -cosec^2(x) or -1/[sin(x)*cos^2(x)].
To find the derivative of cot(x), we can use the quotient rule of differentiation, since cot(x) is formed by dividing cosine and sine functions.
Quotient rule: (f/g)’ = (f’g – g’f)/g^2
So, let f(x) = 1 and g(x) = tan(x)
cot(x) = f(x)/g(x) = 1/tan(x) = cos(x)/sin(x)
Now differentiate cot(x) using the quotient rule:
(cot(x))’ = [(cos(x)/sin(x))’]/(tan^2(x))
= [(cos'(x)sin(x) – cos(x)sin'(x))/sin^2(x)] / (tan^2(x))
= [(-sin(x)sin(x) – cos(x)cos(x))/sin^3(x)] / (tan^2(x))
= -1/[sin(x)*cos^2(x)]
Therefore, the derivative of cot(x) is -cosec^2(x) or -1/[sin(x)*cos^2(x)].
Hence, d/dx cot(x) = -cosec^2(x) or -1/[sin(x)*cos^2(x)].
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