How to Find the Derivative of cot(x) Using the Chain Rule and Trigonometric Identities

d/dx cot(x)

To find the derivative of cot(x) with respect to x, we need to apply the chain rule

To find the derivative of cot(x) with respect to x, we need to apply the chain rule.

The cotangent function can be expressed as cot(x) = cos(x) / sin(x).

Using the quotient rule and the trigonometric identities, we can simplify the derivative as follows:

d/dx [cot(x)] = d/dx [cos(x) / sin(x)]

Before applying the quotient rule, we need to find the derivatives of cos(x) and sin(x):

d/dx [cos(x)] = -sin(x)

d/dx [sin(x)] = cos(x)

Now, using the quotient rule, we can find the derivative of cot(x):

d/dx [cot(x)] = (sin(x) * (-sin(x)) – cos(x) * cos(x)) / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)
= -1 / sin^2(x)
= -csc^2(x)

Therefore, the derivative of cot(x) with respect to x is -csc^2(x), where csc(x) represents the cosecant function.

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