Understanding the Derivative of cot(x) with Respect to x: Step-by-Step Guide and Trigonometric Identities

d/dx cot(x)

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

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

The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:

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

In this case, g(x) = 1 and h(x) = tan(x).

So, g'(x) = 0 (since the derivative of a constant is zero).

To find h'(x), we can use the derivative of the tangent function, which is sec^2(x).

Therefore, h'(x) = sec^2(x).

Now we can substitute these values into the quotient rule formula:

f'(x) = [tan(x) * 0 – 1 * sec^2(x)] / [tan(x)]^2

Simplifying further:

f'(x) = -sec^2(x) / [tan(x)]^2

Now, we want to simplify this expression. To do so, we’ll use the trigonometric identity:

sec^2(x) = 1 + tan^2(x)

Substituting this identity in the expression:

f'(x) = – (1 + tan^2(x)) / [tan(x)]^2

Next, we’ll simplify the expression further:

f'(x) = -1/[tan(x)]^2 – 1

Finally, we can rewrite this in terms of cotangent:

f'(x) = -cot^2(x) – 1

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

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