Understanding How to Find the Derivative of Cot(x) Using the Quotient Rule and Trigonometric Identities

d/dx(cotx)

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

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

Cot(x) can be expressed as the cosine of x divided by the sine of x: cot(x) = cos(x) / sin(x).

Using the quotient rule, the derivative of cot(x) with respect to x is given by:

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

Now, let’s calculate the derivatives of cos(x) and sin(x).

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

(d/dx)(sin(x)) = cos(x)

Plugging these derivatives into our original expression:

(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(x))^2

We can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to simplify the numerator:

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

Therefore, the derivative of cot(x) with respect to x is given by:

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

Note: csc^2(x) is the square of the cosecant function.

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