To find the derivative of the function $y = \cot(x)$, we start with the known result that the derivative is given by

To derive this result, we can apply the quotient rule for differentiation. We rewrite $y = \cot(x)$ as a quotient of two functions. Letting $u = 1$ and $v = \sin(x)$, we have:

Now, we apply the quotient rule, which states that if $y = \frac{u}{v}$, then

Calculating the necessary derivatives, we find:

• $\frac{du}{dx} = 0$ (since $u = 1$),
• $\frac{dv}{dx} = \cos(x)$ (since $v = \sin(x)$).

Substituting these into the quotient rule formula, we get:

This simplifies to:

Next, we can express this derivative in terms of the cotangent and cosecant functions. Recall that:

Using these identities, we can rewrite $y'$:

Thus, we arrive at the conclusion that the derivative of $y = \cot(x)$ is