The Derivative of Cot(x): Explained Using the Quotient Rule and Trigonometric Identities

𝑑/𝑑𝑥[cot 𝑥]

To find the derivative of cot(x), which is the reciprocal of the trigonometric function tangent(x), we can use the quotient rule for differentiation

To find the derivative of cot(x), which is the reciprocal of the trigonometric function tangent(x), we can use the quotient rule for differentiation.

The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, 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, g(x) = 1 (since cot(x) can be written as 1/tan(x)), and h(x) = tan(x).

Now let’s find the derivative of cot(x):

Using the quotient rule, we have:

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

Since g(x) = 1, g'(x) = 0 (since the derivative of a constant is always 0).

So the numerator simplifies to:

[-1 * tan(x)] / (tan(x))^2

Simplifying further, we have:

-tan(x) / tan^2(x)

Using the trigonometric identity tan^2(x) = 1 + cot^2(x), we can rewrite this as:

-tan(x) / (1 + cot^2(x))

Finally, using the reciprocal identity cot(x) = 1/tan(x), we can rewrite this as:

-tan(x) / (1 + (1/tan^2(x)))

Simplifying further, we have:

-tan(x) / (1 + 1/cot^2(x))

Simplifying the denominator, we get:

-tan(x) / (cot^2(x) + 1)

And this can be further simplified to:

-tan(x) / csc^2(x)

So, the derivative of cot(x) with respect to x is:

𝑑/𝑑𝑥[cot 𝑥] = -tan(x) / csc^2(x)

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