cot(ax)の微分

$\cot \left(ax\right)$の微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}x}\cot \left(ax\right)& =& \frac{\mathrm{d}}{\mathrm{d}x}{\{\tan \left(ax\right)\}}^{-1}\dots a,x\in \mathbb{R}\\ & =& \frac{\mathrm{d}}{\mathrm{d}u}{\{\tan \left(u\right)\}}^{-1}\frac{\mathrm{d}u}{\mathrm{d}x}\dots u=ax,\frac{\mathrm{d}u}{\mathrm{d}x}=a\\ & =& \frac{\mathrm{d}}{\mathrm{d}v}{\left\{v\left(u\right)\right\}}^{-1}\frac{\mathrm{d}v}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}\dots v=\tan \left(u\right),\frac{\mathrm{d}v}{\mathrm{d}u}=1+{\tan }^2\left(u\right)\\ & =& -{\left\{v\left(u\right)\right\}}^{-2}\frac{\mathrm{d}v}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}\\ & =& -\frac{1}{{\tan }^2\left(u\right)}\cdot (1+{\tan }^2\left(u\right))\frac{\mathrm{d}u}{\mathrm{d}x}\\ & =& -\frac{1}{{\tan }^2\left(ax\right)}\cdot (1+{\tan }^2\left(ax\right))\cdot a\\ & =& -\frac{a(1+\frac{{\sin }^2\left(ax\right)}{{\cos }^2\left(ax\right)})}{\frac{{\sin }^2\left(ax\right)}{{\cos }^2\left(ax\right)}}\\ & =& -\frac{{\cos }^2\left(ax\right)a(1+\frac{{\sin }^2\left(ax\right)}{{\cos }^2\left(ax\right)})}{{\sin }^2\left(ax\right)}\\ & =& -\frac{a({\cos }^2\left(ax\right)+{\cos }^2\left(ax\right)\frac{{\sin }^2\left(ax\right)}{{\cos }^2\left(ax\right)})}{{\sin }^2\left(ax\right)}\\ & =& -\frac{a({\cos }^2\left(ax\right)+{\sin }^2\left(ax\right))}{{\sin }^2\left(ax\right)}\\ & =& -\frac{a\cdot 1}{{\sin }^2\left(ax\right)}\\ & =& -\frac{a}{{\sin }^2\left(ax\right)}\end{array}$$