定積分の微分

定積分の微分

$$\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}t}\int_0^t g\left(x\right)\mathrm{d}x &=&\frac{\mathrm{d}}{\mathrm{d}t}\left[G\left(x\right)\right]_0^t \\&=&\frac{\mathrm{d}}{\mathrm{d}t}\left[G\left(t\right)-G\left(0\right)\right] \\&=&\frac{\mathrm{d}}{\mathrm{d}t}G\left(t\right)-\frac{\mathrm{d}}{\mathrm{d}t}G\left(0\right) \\&=&g\left(t\right)-0\;\cdots\;0を代入したG\left(0\right)は定数であり，定数の微分は0 \\&=&g\left(t\right) \end{eqnarray}$$