How to Find the Derivative of cot(x) Using the Quotient Rule and Trig Identities

Derivative of cotx

To find the derivative of cot(x), we can use the quotient rule

To find the derivative of cot(x), we can use the quotient rule.

The quotient rule states that for a function u(x) = f(x)/g(x), the derivative is given by:

u'(x) = [f'(x) * g(x) – f(x) * g'(x)] / (g(x))^2

Let’s apply this rule to find the derivative of cot(x).

Given that cot(x) can be written as cos(x)/sin(x), we have:
f(x) = cos(x)
g(x) = sin(x)

First, let’s find f'(x) and g'(x):

f'(x) = -sin(x) (using the derivative of cos(x) which is -sin(x))
g'(x) = cos(x) (using the derivative of sin(x) which is cos(x))

Now, we can substitute these values into the quotient rule formula:

cot'(x) = [f'(x) * g(x) – f(x) * g'(x)] / (g(x))^2
= [-sin(x) * sin(x) – cos(x) * cos(x)] / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)

Recall the trigonometric identity: sin^2(x) + cos^2(x) = 1. We can use this identity to simplify the expression further:

cot'(x) = (-1) / sin^2(x)
= -cosec^2(x)

Therefore, the derivative of cot(x) is -cosec^2(x).

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