Derivative of Cot(x) | Formula and Simplification Using Quotient Rule and Trigonometric Identities

d/dx[cot(x)]

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

To find the derivative of the function cot(x) with respect to x, we can use the quotient rule. The quotient rule states that for a function of the form f(x) = g(x) / h(x), the derivative is given by:

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

In this case, g(x) = 1 and h(x) = tan(x). So, applying the quotient rule, we have:

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

To simplify further, we can use the trigonometric identities: tan(x) = sin(x) / cos(x), cot(x) = cos(x) / sin(x), and sec(x) = 1/ cos(x). Substituting these values, we get:

cot'(x) = [(sin(x) / cos(x)) * (1 / cos(x)) – (cos(x) / sin(x)) * (1 / cos(x))^2 ] / [(sin(x) / cos(x))^2]

cot'(x) = (sin(x) / cos^2(x)) – (cos(x) / sin(x)) / [(sin^2(x) / cos^2(x))]

cot'(x) = sin(x) / cos^2(x) – cos(x) / sin(x) * cos^2(x) / sin^2(x)

cot'(x) = sin(x) / cos^2(x) – cos(x) / (sin(x) * sin(x)) * cos^2(x)

cot'(x) = sin(x) / cos^2(x) – cos(x) * cos^2(x) / (sin(x) * sin^2(x))

Now, let’s simplify further by using the trigonometric identity: sin^2(x) + cos^2(x) = 1.

cot'(x) = sin(x) / cos^2(x) – cos(x) * cos^2(x) / (sin(x) * (1 – cos^2(x)))

cot'(x) = sin(x) / cos^2(x) – cos(x) * cos^2(x) / (sin(x) – sin(x) * cos^2(x))

cot'(x) = [sin(x) – cos(x) * cos^2(x)] / (cos^2(x) * (sin(x) – sin(x) * cos^2(x)))

Finally, we have obtained the derivative of cot(x) with respect to x, which is:

cot'(x) = [sin(x) – cos(x) * cos^2(x)] / (cos^2(x) * (sin(x) – sin(x) * cos^2(x)))

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