Unveiling the Derivative of cot(x) using the Quotient Rule and Trigonometric Identities

d/dx cotx

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 quotient rule states that for two functions u(x) and v(x),

(d/dx) [u(x) / v(x)] = [v(x) * u'(x) – u(x) * v'(x)] / [v(x)]^2.

In this case, let u(x) = 1 and v(x) = tan(x).

Therefore, u'(x) = 0 (since the derivative of a constant is zero),
and v'(x) = sec^2(x) (since the derivative of tan(x) is sec^2(x)).

Applying the quotient rule, we have:

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

Simplifying this expression, we get:

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

Now, we can use the trigonometric identity tan^2(x) + 1 = sec^2(x) to simplify the result:

(d/dx) [cot(x)] = -sec^2(x) / [tan^2(x)] * (tan^2(x) + 1) / (tan^2(x) + 1).

Simplifying further, we get:

(d/dx) [cot(x)] = -1 / (sin^2(x) / cos^2(x) + 1).

Next, we can simplify the denominator:

(d/dx) [cot(x)] = -1 / [(sin^2(x) + cos^2(x)) / cos^2(x)].

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we have:

(d/dx) [cot(x)] = -1 / (1 / cos^2(x)).

Finally, simplifying the expression, we get:

(d/dx) [cot(x)] = -cos^2(x).

Therefore, the derivative of cot(x) is -cos^2(x).

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