Second derivative of $$$\cot^{2}{\left(x \right)}$$$

Find the first derivative $$$\frac{d}{dx} \left(\cot^{2}{\left(x \right)}\right)$$$

The function $$$\cot^{2}{\left(x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = u^{2}$$$ and $$$g{\left(x \right)} = \cot{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\cot^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)}$$

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 2$$$:

$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

Return to the old variable:

$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = 2 {\color{red}\left(\cot{\left(x \right)}\right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

The derivative of the cotangent is $$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:

$$2 \cot{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)} = 2 \cot{\left(x \right)} {\color{red}\left(- \csc^{2}{\left(x \right)}\right)}$$

Thus, $$$\frac{d}{dx} \left(\cot^{2}{\left(x \right)}\right) = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(\cot^{2}{\left(x \right)}\right) = \frac{d}{dx} \left(- 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = -2$$$ and $$$f{\left(x \right)} = \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(- 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dx} \left(\cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)\right)}$$

Apply the product rule $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ with $$$f{\left(x \right)} = \csc^{2}{\left(x \right)}$$$ and $$$g{\left(x \right)} = \cot{\left(x \right)}$$$:

$$- 2 {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)\right)} = - 2 {\color{red}\left(\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right) \cot{\left(x \right)} + \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)}$$

The function $$$\csc^{2}{\left(x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = u^{2}$$$ and $$$g{\left(x \right)} = \csc{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$- 2 \cot{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right)\right)} - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - 2 \cot{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 2$$$:

$$- 2 \cot{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - 2 \cot{\left(x \right)} {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

Return to the old variable:

$$- 4 \cot{\left(x \right)} {\color{red}\left(u\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - 4 \cot{\left(x \right)} {\color{red}\left(\csc{\left(x \right)}\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

The derivative of the cotangent is $$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:

$$- 4 \cot{\left(x \right)} \csc{\left(x \right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)} = - 4 \cot{\left(x \right)} \csc{\left(x \right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) - 2 \csc^{2}{\left(x \right)} {\color{red}\left(- \csc^{2}{\left(x \right)}\right)}$$

The derivative of the cosecant is $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$$- 4 \cot{\left(x \right)} \csc{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} + 2 \csc^{4}{\left(x \right)} = - 4 \cot{\left(x \right)} \csc{\left(x \right)} {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)} + 2 \csc^{4}{\left(x \right)}$$

Simplify:

$$4 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + 2 \csc^{4}{\left(x \right)} = \left(-4 + \frac{6}{\sin^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}$$

Thus, $$$\frac{d}{dx} \left(- 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right) = \left(-4 + \frac{6}{\sin^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\cot^{2}{\left(x \right)}\right) = \left(-4 + \frac{6}{\sin^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}$$$.