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

derivative of cot(x)

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

To find the derivative of 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).

Let’s start by writing cot(x) as a fraction:

cot(x) = 1/tan(x)

Now, let’s differentiate both sides of the equation with respect to x:

d/dx(cot(x)) = d/dx(1/tan(x))

To apply the quotient rule, we can differentiate the numerator and denominator separately. The derivative of 1 is 0, so we only need to focus on the tan(x):

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

Where sec(x) is the secant function, and sec^2(x) is the square of the secant function.

Simplifying further, we have:

d/dx(cot(x)) = -sec^2(x)/(tan^2(x))

Now, we can simplify this expression using trigonometric identities:

The sec^2(x) can be rewritten as 1 + tan^2(x).

d/dx(cot(x)) = -(1 + tan^2(x))/(tan^2(x))

Now, let’s combine the terms:

d/dx(cot(x)) = -1/(tan^2(x)) – tan^2(x)/(tan^2(x))

Simplifying, we obtain:

d/dx(cot(x)) = -1/(tan^2(x)) – 1

Finally, we can further simplify the expression:

d/dx(cot(x)) = -sec^2(x) – 1

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

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