Understanding the Derivative of Cot x and Applying the Quotient Rule

Derivative of cot x

The derivative of cot x can be found by using the quotient rule

The derivative of cot x can be found by using the quotient rule. Let’s first write cot x in terms of sin x and cos x.

cot x = cos x / sin x

Applying the quotient rule, we have:

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

Next, let’s find the derivative of sin x and cos x:

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

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

Substituting these values back into the quotient rule equation, we get:

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

= [(cos^2 x + sin x cos x)] / (sin x)^2

= [(cos^2 x + sin x cos x)] / sin^2 x

= cos x / sin x + cos x / sin^2 x

= cot x + cos x / sin^2 x

Therefore, the derivative of cot x is cot x + cos x / sin^2 x.

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