Exploring the Derivative of Cot(x) and its Application of Quotient Rule in Calculus

(d/dx) cot(x)

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 cotangent function, cot(x), can be expressed as the cosine function divided by the sine function. So, cot(x) = cos(x)/sin(x).

Using the quotient rule, the derivative of cot(x) can be computed as:

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

Now, let’s find the derivatives of cos(x) and sin(x):

(d/dx) cos(x) = -sin(x) (using the derivative of cosine rule)
(d/dx) sin(x) = cos(x) (using the derivative of sine rule)

Substituting these results back into the quotient rule equation:

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

Simplifying further:

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

Recall the trigonometric identity: sin^2(x) + cos^2(x) = 1. We can rewrite -sin^2(x) – cos^2(x) as -1:

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

Finally, using the reciprocal identity: csc^2(x) = 1 / sin^2(x), we can rewrite the expression:

(d/dx) cot(x) = -csc^2(x)

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

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