Finding the Derivative of Cot(x) using the Quotient Rule and Trigonometric Identities

d/dx [cotx]

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

To find the derivative of cot(x), we need to use the quotient rule. The quotient rule states that if we have a function in the form of f(x)/g(x), where f(x) and g(x) are differentiable functions, then the derivative can be found as:

d/dx [f(x)/g(x)] = (g(x) * f'(x) – f(x) * g'(x)) / (g(x))^2

Let’s apply this rule to find the derivative of cot(x):

We can rewrite cot(x) as 1/tan(x), because cot(x) is essentially the reciprocal of tan(x).

So, cot(x) = 1/tan(x)

Now, let’s differentiate both the numerator (1) and denominator (tan(x)) separately:

d/dx [1] = 0 (since 1 is a constant)

d/dx [tan(x)] = sec^2(x) (using the derivative of tan(x) formula)

Now, we can substitute these derivatives into the quotient rule:

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

Simplifying this expression, we have:

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

However, we can simplify this further using trigonometric identities. Recall that sec^2(x) is equivalent to 1 + tan^2(x). Therefore:

d/dx [cot(x)] = – (1 + tan^2(x)) / tan^2(x)

This can also be written as:

d/dx [cot(x)] = – (1/tan^2(x) + 1)

So, the derivative of cot(x) is – (1/tan^2(x) + 1) or alternatively, – (cosec^2(x) + 1).

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