微分の計算問題

微分の計算の動画解説⇒こちらへ

1. 次の関数を微分せよ．

   • ${\displaystyle y=-\frac{1}{x}}$

     ⇒解答

   • ${\displaystyle y=\sqrt{2x+3}}$

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   • ${\displaystyle y={\left(x^3\right)}^{-4}}$

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   • ${\displaystyle y=\left(x+1\right)\left(x^2+3\right)}$

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   • ${\displaystyle y=x^2\left(3-x\right)}$

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   • ${\displaystyle y=\left(2-x\right)\left(2x^2-3x+4\right)}$

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   • ${\displaystyle y=\frac{2x^2-7x+3}{2x-1}}$

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   • ${\displaystyle y=\frac{4x^3-3x+1}{2x+3}}$

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   • ${\displaystyle y=\frac{4x^2-1}{\left(2x+1\right)\left(x-1\right)}}$

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   • ${\displaystyle y=3x^2+2x+1-\frac{3x}{x^2+1}}$

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2. 次の関数を微分せよ．

   • ${\displaystyle y=e^{-x}}$

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   • ${\displaystyle {\displaystyle {y=e}^{3x}}}$

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   • ${\displaystyle y=\sin 6x}$

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   • ${\displaystyle y=\cos 2x}$

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   • ${\displaystyle y=\sqrt[3]{4-x^2}}$

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   • ${\displaystyle y=\sqrt[4]{3x^2-2x+1}}$

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   • ${\displaystyle y=\sqrt[3]{\frac{x+1}{x-1}}}$

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   • ${\displaystyle y=\frac{2x^2+2x+1}{\sqrt{x}}}$

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   • ${\displaystyle y=\frac{x^2+2x+3}{\sqrt{x}}}$

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   • ${\displaystyle y={sin}^3\left(3x+1\right)}$

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   • ${\displaystyle y=sin\left(3x+2\right)}$

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   • ${\displaystyle y=cos\left(5x-2\right)}$

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   • ${\displaystyle y=cos3x-sin\left(-2x+1\right)}$

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   • ${\displaystyle y=\tan 2x}$

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   • ${\displaystyle y=tan\frac{1}{2x}}$

     ⇒解答
3. 次の関数を微分せよ．

   • ${\displaystyle y={cos}^4\left(3x-2\right)}$

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   • ${\displaystyle y={tan}^3\left(4x-1\right)}$

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   • ${\displaystyle y=sin\left(4x-1\right)cos3x}$

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   • ${\displaystyle y={\log }_{x}3}$

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   • ${\displaystyle y=\frac{1-3sinx}{2cosx}}$

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   • ${\displaystyle y=\left(3x^2-1\right)tan\frac{1}{2x}}$

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   • ${\displaystyle y=log\left(2x^2-3x+2\right)}$

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   • ${\displaystyle y=4^x}$

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4. 次の関数を微分せよ．

   • ${\displaystyle y={\sin }^{-1}2x}$

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   • ${\displaystyle y=5-9x+\frac{9}{2}{x}^2}$

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   • ${\displaystyle x=t^2+\frac{6}{t^3}}$

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   • ${\displaystyle y=\frac{t^5+1}{t^5}}$

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   • ${\displaystyle f\left(u\right)=u^2+6\sqrt[3]{u^2}}$

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   • ${\displaystyle y=\log \left(\log \left(\log \left(\log 5x\right)\right)\right)}$

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   • ${\displaystyle y=x^{5x}}$

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   • ${\displaystyle y=e^{-3x}\sin 3x}$

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   • ${\displaystyle y=\frac{{\left(5x+1\right)}^5}{{\left(x^3-4\right)}^4}}$

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   • ${\displaystyle y=\sqrt[5]{5x-7}}$

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   • ${\displaystyle y=x^{\frac{1}{3n}}}$

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   • ${\displaystyle y={\sin }^{3}4x{\cos }^{4}3x}$

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   • ${\displaystyle y=\log \left(\sin x\right)}$

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   • ${\displaystyle y=\frac{\sin 6x}{\cos 2x}}$

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   • ${\displaystyle y=\frac{e^6\cos 6x}{\log 3x}}$

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   • ${\displaystyle y=\log \left|\frac{1-2x}{1+2x}\right|}$

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   • ${\displaystyle y=\log \frac{e^{3x}-3}{e^{3x}+1}}$

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   • ${\displaystyle y=\frac{e^x-e^{-x}}{e^x+e^{-x}}}$

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   • ${\displaystyle y=\frac{1}{n}\log \frac{e^{-2nx}+e^{2nx}}{4}}$

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   • ${\displaystyle y=\frac{\sin 2x-4\cos x}{2\sin x\cos x}}$

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   • ${\displaystyle y={\sin }^{-1}\frac{x}{\sqrt{7}}}$

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   • ${\displaystyle y={\cos }^{-1}\sqrt{2x}}$

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   • ${\displaystyle y={\cos }^{-1}\frac{2}{x}}$

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   • ${\displaystyle y={\tan }^{-1}6x}$

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   • ${\displaystyle y={\tan }^{-1}\frac{13}{x}}$

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   • ${\displaystyle y=\log \left(x+\sqrt{x^2+4}\right)}$

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   • ${\displaystyle y=\log \left(\cos 3x\right)}$

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   • ${\displaystyle y=\log \left(\tan \frac{x}{2}\right)}$

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   • ${\displaystyle y=7^{x^3-x}}$

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   • ${\displaystyle y=3\cdot 5{}^{x^4-3x}}$

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   • ${\displaystyle y={\sin }^{5}6x}$

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   • ${\displaystyle y=\text{}\frac{4x^2-5}{\cos 7x}}$

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   • ${\displaystyle y=\frac{\cos 7x}{\sin 4x}}$

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   • ${\displaystyle y=\sqrt{1+\sin 3x}}$

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   • ${\displaystyle y=\frac{x}{x+\sqrt{x^2+1}}}$

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   • ${\displaystyle y=\log \left(\sin 3x\right)}$

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   • ${\displaystyle y=x^x}$　 (${\displaystyle x>0}$ )

     ⇒解答
5. 次の関数の第2次導関数を求めなさい．

   • ${\displaystyle y=x^4+3x^2-7x+9}$

     ⇒解答

   • ${\displaystyle y=e^{3x}\cos 6x}$

     ⇒解答

   • ${\displaystyle y=\log \left(\sin 5x\right)}$

     ⇒解答

   • ${\displaystyle y=e^{ax}\cos bx}$

     ⇒解答

   • ${\displaystyle y=-5t^5+6t^4+3t^3+7}$

     ⇒解答

   • ${\displaystyle y=6\sin 4x}$

     ⇒解答

   • ${\displaystyle y=7\cos 5x^2}$

     ⇒解答

   • ${\displaystyle y=3x^3\sin 4x^2}$

     ⇒解答

   • ${\displaystyle y=4\left(3-e^{-7x}\right)}$

     ⇒解答

   • ${\displaystyle y=5\left(4+e^{x^3-2x}\right)}$

     ⇒解答

   • ${\displaystyle y=\log \left(4x-6\right)}$

     ⇒解答

   • ${\displaystyle y=2\log \left(3x^2-1\right)}$

     ⇒解答

   • ${\displaystyle y=3\tan 2x}$

     ⇒解答
6. 次の条件を満たす時，3次関数を求めなさい．

   • ${\displaystyle x{f}^{″}\left(x\right)+\left(3+x\right)f^{\prime }\left(x\right)-3f\left(x\right)}$ ${\displaystyle =0}$，${\displaystyle f\left(0\right)=1}$　⇒解答

   • ${\displaystyle f\left(0\right)=4,f\left(1\right)=10,}$${\displaystyle f^{\prime }\left(2\right)=23,}$ ${\displaystyle f^{″}\left(3\right)=22}$　　⇒解答

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7. 次の問題を解きなさい

   • 	次の条件式を満たす${\displaystyle a,b}$を求めよ．
     	${\displaystyle y=e^x+e^{-2x}}$，${\displaystyle y^{″}+3a{y}^{\prime }+by=0}$

   • ⇒解答

   • 	次の関数の${\displaystyle y^{″}}$を ${\displaystyle x}$を用いずに，${\displaystyle y,y^{\prime }}$ を用いて表せ．
     	${\displaystyle y=e^{ax}\cos bx}$

   • ⇒解答

   • 	${\displaystyle f\left(0\right)=g\left(0\right),f^{\prime }\left(0\right)=g^{\prime }\left(0\right),}$ ${\displaystyle f^{″}\left(0\right)=g^{″}\left(0\right)}$ を満たす${\displaystyle a,b,c}$ を求めよ．
     	${\displaystyle f\left(x\right)=3\sin x+2}$,${\displaystyle g\left(x\right)=\frac{a}{b{x}^2+cx+3}}$

   • ⇒解答

   • 	${\displaystyle f\left(x\right)={\log }_{2}x}$ とする．${\displaystyle f^{\prime }\left(5\right)}$を微分係数の定義式を用いて求めよ．
     	微分係数の定義式：${\displaystyle f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}}$

   • ⇒解答

8. 媒介変数(パラメータ)表示された関数 ${\displaystyle x=\sqrt{t-1}-2}$ ， ${\displaystyle y=t^3-5}$ について導関数 ${\displaystyle \frac{dy}{dx}}$ を $t$ の式で表し，点 ${\displaystyle \text{P}\left(-1,3\right)}$ における接線方程式を求めよ．　　⇒解答

9. 陰関数の微分法を用いて曲線 ${\displaystyle 4x^2+9y^2-36y=0}$ の ${\displaystyle \frac{dy}{dx}}$ をもとめ，曲線上の点 ${\displaystyle \left(\frac{3\sqrt{3}}{2},1\right)}$ における接線方程式を求めよ．　　⇒解答

 

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2024年9月24日

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