導関数の基本式 I（微分の公式I）

• ${\displaystyle {\left\{c\right\}}^{\prime }=0}$

  すなわち

  $f\left(x\right)=c\to f^{\prime }\left(x\right)=0$     ⇒導出計算

• ${\displaystyle {\left\{cg\left(x\right)\right\}}^{\prime }=c{g}^{\prime }\left(x\right)}$

  すなわち

  $f\left(x\right)=cg\left(x\right)\to f^{\prime }\left(x\right)=c{g}^{\prime }\left(x\right)$     ⇒導出計算

• ${\displaystyle {\left\{g\left(x\right)\pm h\left(x\right)\right\}}^{\prime }=g^{\prime }\left(x\right)\pm h^{\prime }\left(x\right)}$

  すなわち

  $f\left(x\right)=g\left(x\right)\pm h\left(x\right)$ $\to f^{\prime }\left(x\right)=g^{\prime }\left(x\right)\pm h^{\prime }\left(x\right)$     ⇒導出計算

• ${\displaystyle {\left\{g\left(x\right)h\left(x\right)\right\}}^{\prime }}$ ${\displaystyle =g^{\prime }\left(x\right)h\left(x\right)+g\left(x\right)h^{\prime }\left(x\right)}$

  すなわち

  ${\displaystyle f\left(x\right)=g\left(x\right)h\left(x\right)}$ ${\displaystyle \to f^{\prime }\left(x\right)=g^{\prime }\left(x\right)h\left(x\right)+g\left(x\right)h{\left(x\right)}^{\prime }}$     ⇒導出計算

• ${\displaystyle {\left\{\frac{1}{g\left(x\right)}\right\}}^{\prime }=-\frac{g^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^2}}$

  すなわち

  ${\displaystyle f\left(x\right)=\frac{1}{g\left(x\right)}\to f^{\prime }\left(x\right)=-\frac{g^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^2}}$       ⇒導出計算

• ${\displaystyle {\left\{\frac{h\left(x\right)}{g\left(x\right)}\right\}}^{\prime }=\frac{h^{\prime }\left(x\right)g\left(x\right)-h\left(x\right)g^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^2}}$

  すなわち

  ${\displaystyle f\left(x\right)=\frac{h\left(x\right)}{g\left(x\right)}}$ ${\displaystyle \to f^{\prime }\left(x\right)=\frac{h^{\prime }\left(x\right)g\left(x\right)-h\left(x\right)g^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^2}}$     ⇒導出計算

 

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最終更新日： 2025年4月25日