A Guide to Finding the Derivative of the Cotangent Function Using the Quotient Rule and Simplification

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To find the derivative of the function f(x) = cot(x) with respect to x, we’ll use the quotient rule. The quotient rule states that if we have a function u(x) = f(x)/g(x), then the derivative of u(x) with respect to x is given by:

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

In our case, f(x) = 1 and g(x) = tan(x). Differentiating these functions, we have:

f'(x) = 0 (since the derivative of any constant is zero)
g'(x) = sec^2(x) (since the derivative of tan(x) is sec^2(x))

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

u'(x) = (0 * tan(x) – 1 * sec^2(x)) / [tan(x)]^2

Simplifying further, we have:

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

However, we can express cot(x) in terms of sin(x) and cos(x) to further simplify the derivative. Recall that cot(x) = cos(x) / sin(x).

Substituting these values into u'(x), we have:

u'(x) = -sec^2(x) / [tan(x)]^2 = -1 / [sin^2(x) / cos^2(x)] = -cos^2(x) / sin^2(x)

Therefore, the derivative of cot(x) is -cos^2(x) / sin^2(x).

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