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

The calculator will find the second derivative of $\csc^{2}{\left(x \right)}$, with steps shown.

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Find $\frac{d^{2}}{dx^{2}} \left(\csc^{2}{\left(x \right)}\right)$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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