Derivative of cot(x) | Applying the quotient rule and trigonometric identities to find the derivative of cot(x)

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 cosine of x divided by the sine of x:

cot(x) = cos(x) / sin(x)

Now, let’s differentiate both the numerator and denominator separately:

d/dx [cos(x)] = -sin(x)
d/dx [sin(x)] = cos(x)

Now, applying the quotient rule:

d/dx [cot(x)] = (sin(x) * -sin(x) – cos(x) * cos(x)) / (sin(x))^2

Simplifying further:

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

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

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

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

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