合成関数の微分の例

[例1]

$$y = e^{ax}$$

$t=ax$と置いて、

$$\begin{eqnarray*} \frac{dy}{dx}&=&\frac{dy}{dt}\cdot\frac{dt}{dx}\\ &=&\left(... ...ot\left(\frac{d}{dx}ax\right)\\ &=&e^t\times a\\ &=& ae^{ax} \end{eqnarray*}$$

[例2]

$$y = \sin^2 x$$

$t=\sin x$と置くと、

$$\begin{eqnarray*} y&=&t^2\\ \frac{dy}{dx}&=&\frac{dt^2}{dx} =\frac{dt^2}{dt}... ...imes(\sin x)'\\ &=&2t\times\cos x\\ &=& 2\sin x\times \cos x \end{eqnarray*}$$

[例3]

$$\begin{eqnarray*} y &=& \cos x\\ \frac{dy}{dx}&=&\frac{d}{dx}\cos x =\frac{d... ...^{jx}-e^{-jx})\\ &=&\frac{j}{2}\times 2j\sin x\\ &=& -\sin x \end{eqnarray*}$$

[例4]

$$y = x^p$$

$t=\log y$と置き、上式の両辺の対数を計算すると、

$$t = \log y = \log x^p = p\log x$$

$$\begin{eqnarray*} \frac{dt}{dy}&=&(\log y)'=\frac{1}{y}\\ \frac{dt}{dx}&=&(p\... ...=&\frac{p/x}{1/y}= \frac{py}{x} =\frac{px^p}{x}\\ &=&px^{p-1} \end{eqnarray*}$$