The Derivative of Cot(x): Applying the Quotient Rule and Simplifying

d/dx(cot(x))

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

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

The cotangent function can be expressed as the ratio of cosine to sine: cot(x) = cos(x) / sin(x).

Let’s differentiate both the numerator and denominator separately:

Numerator:
The derivative of cos(x) with respect to x is -sin(x).

Denominator:
The derivative of sin(x) with respect to x is cos(x).

Now, apply the quotient rule:

d/dx(cot(x)) = (denominator * numerator’ – numerator * denominator’) / (denominator)^2

= (sin(x) * (-sin(x)) – cos(x) * cos(x)) / (sin(x))^2

= (-sin^2(x) – cos^2(x)) / sin^2(x)

Using the trigonometric identity that sin^2(x) + cos^2(x) = 1, we can simplify further:

= -1 / sin^2(x)

Therefore, the derivative of cot(x) with respect to x is -1 / sin^2(x) or -csc^2(x) (cosec^2(x)).

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