Derivative of Cot(x) Formula and Simplification | Quotient Rule and Chain Rule Application

d/dx[cotx]

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

To find the derivative of the function f(x) = cot(x), we can use the quotient rule. The cotangent function is defined as the reciprocal of the tangent function, cot(x) = 1/tan(x).

Using the quotient rule, we have:

f(x) = 1/tan(x)

Using the chain rule, we know that the derivative of tan(x) is sec^2(x):

f'(x) = [1′(x) * tan(x) – 1(x) * tan'(x)] / (tan^2(x))

Now simplifying this expression, we find:

f'(x) = [-sec^2(x) * 1 – 1 * sec^2(x)] / (tan^2(x))

f'(x) = [-sec^2(x) – sec^2(x)] / (tan^2(x))

f'(x) = -2sec^2(x) / tan^2(x)

Now, we can use the identity 1 + tan^2(x) = sec^2(x) to rewrite this in a more simplified form:

f'(x) = -2sec^2(x) / (sec^2(x) – 1)

f'(x) = -2 / (sec^2(x) – 1)

Therefore, the derivative of cot(x) with respect to x is -2 / (sec^2(x) – 1).

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