d/dx [cot x]=
-csc^2 x
We can use the quotient rule and the chain rule to find the derivative of cot x:
Let y = cot x.
Then, y = cos x/sin x.
Applying the quotient rule:
dy/dx = [(sin x)(-sin x) – (cos x)(cos x)] / (sin x)^2
dy/dx = [-sin^2 x – cos^2 x] / sin^2 x
dy/dx = -1/sin^2 x
But we know that sin^2 x + cos^2 x = 1.
Therefore, sin^2 x = 1 – cos^2 x.
Substituting back in the derivative:
dy/dx = -1/[(1 – cos^2 x)]
dy/dx = -cos^2 x / [(1 – cos^2 x)^2]
Thus, the derivative of cot x is -cos^2 x / [(1 – cos^2 x)^2].
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