Understanding the Derivative of Cot(x): A Step-by-Step Guide

d/dx [ cot(x) ]

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

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

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

In this case, we have f(x) = cot(x), which can be written as f(x) = cos(x)/sin(x). Therefore, g(x) = cos(x) and h(x) = sin(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = -sin(x) (derivative of cos(x))
h'(x) = cos(x) (derivative of sin(x))

Now let’s apply the quotient rule:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
= ((-sin(x)) * sin(x) – cos(x) * cos(x)) / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)

Now we can simplify the expression:

Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we have:

f'(x) = (-1) / sin^2(x)

Recall that cot(x) is defined as cos(x)/sin(x). Therefore, we can rewrite f'(x) as:

f'(x) = (-1) / sin^2(x) = -csc^2(x)

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

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