What is the derivative of cot(x)?

To find the derivative of cot(x), we can use the fact that cot(x) can be expressed as the ratio of cosine and sine:

[ \text{cot}(x) = \frac{\cos(x)}{\sin(x)} ]

Using the quotient rule for differentiation, which states that if you have a function that is the quotient of two functions ( u(x) ) and ( v(x) ), the derivative ( (u/v)' ) is given by:

[ \frac{u'v - uv'}{v^2} ]

In this case, let ( u = \cos(x) ) and ( v = \sin(x) ). The derivatives are ( u' = -\sin(x) ) and ( v' = \cos(x) ). Plugging these into the quotient rule gives:

[

\frac{d}{dx} \text{cot}(x) = \frac{-\sin(x)\sin(x) - \cos(x)\cos(x)}{\sin^2(x)}

]

This simplifies to:

[ \frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)} ]

Using the Pythag