Derivative of Cot(x) | Using the Quotient Rule to Find the Derivative of Cot(x)

Derivative of cot(x)

To find the derivative of cot(x), we will use the quotient rule, since cot(x) can be written as the ratio of two functions

To find the derivative of cot(x), we will use the quotient rule, since cot(x) can be written as the ratio of two functions.

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 the case of cot(x), we can rewrite cot(x) as 1/tan(x).
So, cot(x) = 1/tan(x)

Now, we differentiate cot(x) using the quotient rule.
Let’s set g(x) = 1 and h(x) = tan(x).

g'(x) = 0 (since 1 is a constant) and h'(x) = sec^2(x) (derivative of tan(x))

Using the quotient rule formula, we have:

cot'(x) = (0*tan(x) – 1*sec^2(x))/(tan(x))^2

Simplifying the equation further, we get:

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

Since sec^2(x) = 1 + tan^2(x), we can substitute that into the equation:

cot'(x) = -1/(tan(x))^2 * (1 + tan^2(x))

Simplifying the equation even more, we have:

cot'(x) = -1/(tan(x))^2 – 1

Therefore, the derivative of cot(x) is -1/(tan(x))^2 – 1, which can also be written as -csc^2(x) – 1.

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