d/dx cot x
-csc^2 x
To find the derivative of cot x with respect to x, we’ll need to use the quotient rule of differentiation since cot x is expressed as a quotient of cos x and sin x:
d/dx cot x = d/dx (cos x / sin x)
Using the quotient rule, we can express the derivative as:
= (sin x d/dx(cos x) – cos x d/dx(sin x)) / sin^2 x
To simplify this expression, we can use the following trigonometric identities:
d/dx(cos x) = -sin x
d/dx(sin x) = cos x
Substituting these values into the derivative expression, we get:
= (sin x (-sin x) – cos x (cos x)) / sin^2 x
= (-sin^2 x – cos^2 x) / sin^2 x
= -1/(sin x cos x)
Therefore, the derivative of cot x with respect to x is -1/(sin x cos x).
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