d/dx (cot x)
-csc^2 x
We start by rewriting cot(x) in terms of sine and cosine:
cot(x) = cos(x)/sin(x)
Now, we apply the quotient rule of differentiation:
d/dx (cot(x)) = [d/dx(cos(x)) * sin(x) – cos(x) * d/dx(sin(x))] / [sin^2(x)]
The derivatives of sine and cosine are:
d/dx(cos(x)) = -sin(x)
d/dx(sin(x)) = cos(x)
Now, we substitute these derivatives back into the original equation:
d/dx(cot(x)) = [-sin(x) * sin(x) – cos(x) * cos(x)] / [sin^2(x)]
= -[1 + tan^2(x)]/sin^2(x)
Therefore, the derivative of cot(x) is -[1 + tan^2(x)]/sin^2(x).
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