How to Find the Derivative of cot(x) and its Simplified Form

Derivative of cot x

To find the derivative of cot(x), we can start by using the quotient rule

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

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

In this case, our function is cot(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))

Substituting these values into the quotient rule, we have:

cot'(x) = ((-sin(x))(sin(x)) – (cos(x))(cos(x)))/((sin(x))^2)

Simplifying, we get:

cot'(x) = (-sin^2(x) – cos^2(x))/(sin^2(x))

Now, using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can rewrite the numerator:

(-sin^2(x) – cos^2(x)) = -1

Therefore, the derivative of cot(x) is:

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

So, the derivative of cot(x) is -1/(sin^2(x)).

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