\( 2.1 \quad y=\cot ^{2} 5 x \)

We start with the function \[ y = \cot^2(5x). \] This can be seen as a composition of functions. Let \[ u = \cot(5x) \] so that \[ y = u^2. \] **Step 1. Differentiate \(y\) with respect to \(u\).** Since \[ y = u^2, \] its derivative with respect to \(u\) is \[ \frac{dy}{du} = 2u. \] **Step 2. Differentiate \(u\) with respect to \(x\).** We have \[ u = \cot(5x). \] Recall that the derivative of \(\cot u\) with respect to \(u\) is \[ \frac{d}{du}[\cot u] = -\csc^2 u. \] Since \(u = 5x\), by the chain rule, we also differentiate the inner function: \[ \frac{d}{dx}(5x) = 5. \] Thus, \[ \frac{du}{dx} = \frac{d}{dx}[\cot(5x)] = -\csc^2(5x) \cdot 5 = -5\csc^2(5x). \] **Step 3. Apply the Chain Rule.** Using the chain rule, \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}, \] substitute the expressions we found: \[ \frac{dy}{dx} = 2u \cdot \left(-5\csc^2(5x)\right). \] Substitute back \(u = \cot(5x)\): \[ \frac{dy}{dx} = 2\cot(5x) \cdot \left(-5\csc^2(5x)\right). \] **Step 4. Simplify the expression.** Multiplying the constants, \[ \frac{dy}{dx} = -10\cot(5x)\csc^2(5x). \] Thus, the derivative of the function is \[ \boxed{\frac{dy}{dx} = -10\cot(5x)\csc^2(5x)}. \]