導関数の計算と性質
【導関数の計算】
(1)\((x^n)'=nx^{n-1}\)
(2)\((c)'=0\)
【導関数の性質】
(1)\(\{kf(x)\}'=kf'(x)\)
(2)\(\{f(x)+g(x)\}'=f'(x)+g'(x)\)
(3)\(\{f(x)-g(x)\}'=f'(x)-g'(x)\)
(4)\(\{f(x)g(x)\}'=f'(x)g(x)+f(x)g'(x)\)
(5)\(\displaystyle \left\{\frac{f(x)}{g(x)}\right\}'=\frac{f'(x)g(x)-f(x)g'(x)}{\{g(x)\}^2}\)
【例題】次の関数を微分しなさい。
(1)\(y=(x+3)(2x^2-1)\)
\(y'=(x+3)'(2x^2-1)+(x+3)(2x^2-1)'\)
\(=1(2x^2-1)+(x+3)4x\)
\(=2x^2-1+4x^2+12x\)
\(=6x^2+12x-1\)
(2)\(\displaystyle y=\frac{1}{x+1}\)
\(\displaystyle y'=-\frac{(x+1)'}{(x+1)^2}\)
\(\displaystyle =-\frac{1}{(x+1)^2}\)
(3)\(\displaystyle y=\frac{2}{x}\)
\(\displaystyle y'=-\frac{2(x)'}{x^2}\)
\(\displaystyle =-\frac{2}{x^2}\)
(4)\(\displaystyle y=\frac{2x}{x^2-1}\)
\(\displaystyle y'=\frac{(2x)'(x^2-1)-2x(x^2-1)'}{(x^2-1)^2}\)
\(\displaystyle =\frac{2(x^2-1)-2x・2x}{(x^2-1)^2}\)
\(\displaystyle =-\frac{2(x^2+1)}{(x^2-1)^2}\)