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関数の積と商の微分法の問題の解法 | 夢を叶える塾
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関数の積と商の微分法の問題の解法

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関数の積と商の微分法を使って導関数を求める問題です。

基本問題

次の関数の導関数を求めなさい。
(1)y=(2x+1)(3x^2+5x+1)
(2)\displaystyle y=\frac{x^3+1}{3x+1}

解き方

積と商の微分法を使って微分していきます。
積の微分法
uv=u'v+uv'
商の微分法
\displaystyle \frac{u}{v}=\frac{u'v-uv'}{v^2}
ですね。
ちなみに商の微分法は覚え無くても積の微分法と合成関数の微分法を使って導けますね。
\displaystyle uv^{-1}=u'v^{-1}+u(v^{-1})'=u'v^{-1}-u(-v^{-2})v'=\frac{u'v-uv'}{v^2}
w=v^{-1}と置くと少しわかりやすくなるかもしれません。
\displaystyle \frac{d(uv^{-1})}{dx}=\frac{d(uw)}{dx}=\frac{d(u)}{dx}w+\frac{d(w)}{dx}u=u'w+\frac{d(w)}{dv}\frac{dv}{dx}u=u'w+u(-v^{-2})v'=u'v^{-1}-uv^{-2}v'=\frac{u'v-uv'}{v^2}

解説

(1)y=(2x+1)(3x^2+5x+1)
積の微分法を使います。
y'=((2x+1)(3x^2+5x+1))'=(2x+1)'(3x^2+5x+1)+(2x+1)(3x^2+5x+1)'
=2(3x^2+5x+1)+(2x+1)(6x+5)=6x^2+10x+2+12x^2+16x+5=18x^2+26x+7

(2)\displaystyle y=\frac{x^3+1}{3x+1}
商の微分法を使います。
\displaystyle y'=\left(\frac{x^3+1}{3x+1}\right)'=\frac{(x^3+1)'(3x+1)-(x^3+1)(3x+1)'}{(3x+1)^2}
\displaystyle =\frac{(3x^2)(3x+1)-(x^3+1)\cdot 3}{(3x+1)^2}=\frac{9x^3+3x^2-3x^3-3}{(3x+1)^2}
\displaystyle =3\frac{2x^3+x^2-1}{(3x+1)^2}

終わりに

合成関数の微分法程ではないかもしれませんが、この後の三角関数や指数関数、対数関数の微分法で必要になります。
数Ⅲの微分の基礎です。
合成関数の微分法同様、数Ⅱの段階で知っているとだいぶ楽できると思います。

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