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# 数学演習/数学III/微分法

## 微分法

### 微分係数と導関数

〔1〕　次の関数${\displaystyle f(x)}$の導関数を定義に従って求めよ。

(1) ${\displaystyle f(x)=x^{2}+1}$
(2) ${\displaystyle f(x)={\frac {2}{x}}}$
(3) ${\displaystyle f(x)={\sqrt {x}}}$

〔2〕　関数${\displaystyle f(x)=|x+1|}$${\displaystyle x=-1}$で微分可能でないことを示せ。

### 微分の計算の基本

(1) ${\displaystyle f(x)=x^{4}+2x^{3}-5x^{2}+3x+2}$
(2) ${\displaystyle f(x)=(x^{2}+3)(4x+2)}$
(3) ${\displaystyle f(x)={\frac {3x^{2}+1}{x+2}}}$
(4) ${\displaystyle f(x)=(2x+7)^{5}}$
(5) ${\displaystyle f(x)={\frac {1}{(2x+7)^{5}}}}$

### 色々な微分の計算

(1) ${\displaystyle f(x)=\sin(x^{2}-3)}$
(2) ${\displaystyle f(x)=\cos({\sqrt {4x-3}})}$
(3) ${\displaystyle f(x)=\sin \left(x+{\frac {\pi }{3}}\right)}$
(4) ${\displaystyle f(x)=\log {x^{2}}}$
(5) ${\displaystyle f(x)=\log {(\sin x)}}$
(6) ${\displaystyle f(x)={\frac {(x-3)^{5}}{\sqrt[{3}]{2x+5}}}}$
(7) ${\displaystyle f(x)=e^{2x+1}}$
(8) ${\displaystyle f(x)=2^{x^{2}}}$
(9) ${\displaystyle f(x)={\frac {x^{2}}{\log {x}}}}$
(10) ${\displaystyle f(x)=\log {\sqrt {\frac {2x^{2}-1}{2x+1}}}}$

### 第n次導関数

〔1〕　次の関数${\displaystyle f(x)}$を第3次までの導関数を求めよ。

(1) ${\displaystyle f(x)=5x^{4}-4x^{3}+2x^{2}}$
(2) ${\displaystyle f(x)=x\sin x}$
(3)${\displaystyle f(x)={\frac {x^{2}+1}{x+1}}}$

### 陰関数の導関数

(1) ${\displaystyle y^{2}=4x}$
(2) ${\displaystyle {\frac {x^{2}}{16}}+{\frac {y^{2}}{9}}=1}$
(3) ${\displaystyle x^{2}+4xy+y^{2}=0}$