Segunda derivada de $$$\csc^{2}{\left(x \right)}$$$

Encontre a primeira derivada $$$\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right)$$$

A função $$$\csc^{2}{\left(x \right)}$$$ é a composição $$$f{\left(g{\left(x \right)} \right)}$$$ de duas funções $$$f{\left(u \right)} = u^{2}$$$ e $$$g{\left(x \right)} = \csc{\left(x \right)}$$$.

Aplique a regra da cadeia $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

Aplique a regra de poder $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ com $$$n = 2$$$:

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

Volte para a variável antiga:

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

A derivada da cossecante é $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

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

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

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

Aplique a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = -2$$$ e $$$f{\left(x \right)} = \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$$:

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

Aplique a regra do produto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ com $$$f{\left(x \right)} = \csc^{2}{\left(x \right)}$$$ e $$$g{\left(x \right)} = \cot{\left(x \right)}$$$:

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

A função $$$\csc^{2}{\left(x \right)}$$$ é a composição $$$f{\left(g{\left(x \right)} \right)}$$$ de duas funções $$$f{\left(u \right)} = u^{2}$$$ e $$$g{\left(x \right)} = \csc{\left(x \right)}$$$.

Aplique a regra da cadeia $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

Aplique a regra de poder $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ com $$$n = 2$$$:

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

Volte para a variável antiga:

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

A derivada da cossecante é $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

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

A derivada da cotangente é $$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:

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

Simplificar:

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

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

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