Derivative of cot(x): Exploring Quotient Rule & Alternative Expressions

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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 can be written as cos(x)/sin(x), so we have:

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

Now, let’s apply the quotient rule:

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

To find (d/dx)(cos(x)), we differentiate cos(x) with respect to x.
(d/dx)(cos(x)) = -sin(x)

To find (d/dx)(sin(x)), we differentiate sin(x) with respect to x.
(d/dx)(sin(x)) = cos(x)

Substituting these derivatives back into the quotient rule equation:

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

Simplifying the equation:

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

Recall that sin^2(x) = 1 – cos^2(x), so we can rewrite the derivative as:

(d/dx)(cot(x)) = -1/(1 – cos^2(x))

Alternatively, we can rewrite the cotangent function as a ratio of sine and cosine:

cot(x) = cos(x)/sin(x) = 1/(tan(x))

Taking the derivative of 1/(tan(x)) is straightforward as it involves the chain rule:

(d/dx)[1/(tan(x))] = -sec^2(x)

So, both -1/(1 – cos^2(x)) and -sec^2(x) are valid expressions for the derivative of cot(x).

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