数学 - 解析学 - 微分と基本的な関数 - 微分係数、導関数 - 合成微分律(合成関数の微分法)

解析入門 原書第3版
原著

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解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第3章(微分係数、導関数)、6(合成微分律(合成関数の微分法))、1-40.を取り組んでみる。

1-40.

1. 1. 
      $f\left(u\right)=u^8,g\left(x\right)=x+1$
   2. 
      $f\left(u\right)=u^{\frac{1}{2}},g\left(x\right)=2x-5$
   3. 
      $f\left(u\right)=u^3,g\left(x\right)=\sin x$
   4. 
      $f\left(u\right)=u^5,g\left(x\right)=\log x$
   5. 
      $f\left(u\right)=\sin u,g\left(x\right)=2x$
   6. 
      $f\left(u\right)=\log u,g\left(x\right)=x^2+1$
   7. 
      $f\left(u\right)=e^u,g\left(x\right)=\cos x$
   8. 
      $f\left(u\right)=\log u,g\left(x\right)=e^x+\sin x$
   9. 
      $f\left(u\right)=\sin u,g\left(x\right)=\log x+\frac{1}{x}$
   10. 
       $f\left(u\right)=\frac{\frac{1}{2}u+1}{\sin u},g\left(x\right)=2x$
   11. 
       $f\left(u\right)=u^3,g\left(x\right)=2x^2+3$
   12. 
       $f\left(u\right)=\cos u,g\left(x\right)=\sin 5x$
   13. 
       $f\left(u\right)=\log u,g\left(x\right)=\cos 2x$
   14. 
       $f\left(u\right)=\sin u,g\left(x\right)={\left(2x+5\right)}^2$
   15. 
       $f\left(u\right)=\sin u,g\left(x\right)=\cos \left(x+1\right)$
   16. 
       $f\left(u\right)=\sin u,g\left(x\right)=e^x$
   17. 
       $f\left(u\right)=\frac{1}{u^4},g\left(x\right)=3x-1$
   18. 
       $f\left(u\right)=\frac{1}{u^3},g\left(x\right)=4x$
   19. 
       $f\left(u\right)=\frac{1}{u^2},g\left(x\right)=\sin 2x$
   20. 
       $f\left(u\right)=\frac{1}{u^2},g\left(x\right)=\cos 2x$
   21. 
       $f\left(u\right)=\frac{1}{\sin u},g\left(x\right)=3x$
   22. 
       $f\left(u\right)=\frac{1}{\frac{1}{2}u},g\left(x\right)=2\sin x\cos x=\sin 2x$
   23. 
       $f\left(u\right)=\left(u^2+1\right)e^u,g\left(x\right)=x$
   24. 
       $f\left(u\right)=\left(u^3+2u\right)\left(\sin 3u\right),g\left(x\right)=x$
   25. 
       $f\left(u\right)=\frac{1}{u},g\left(x\right)=\sin x+\cos x$
   26. 
       $f\left(u\right)=\frac{\sin 2u}{e^u},g\left(x\right)=x$
   27. 
       $f\left(u\right)=\frac{\log u}{u^2+3},g\left(x\right)=x$
   28. 
       $f\left(u\right)=\frac{u+1}{\cos 2u},g\left(x\right)=x$
   29. 
       $f\left(u\right)=\left(2u-3\right)\left(e^u+u\right),g\left(x\right)=x$
   30. 
       $f\left(u\right)=u,g\left(x\right)=\left(x^3-1\right)\left(e^{3x}+5x\right)$
   31. 
       $f\left(u\right)=u,g\left(x\right)=\frac{x^3+1}{x-1}$
   32. 
       $f\left(u\right)=u,g\left(x\right)=\frac{x^2-1}{2x+3}$
   33. 
       $f\left(u\right)=u,g\left(x\right)=\left(x^{\frac{4}{3}}-e^x\right)\left(2x+1\right)$
   34. 
       $f\left(u\right)=u,g\left(x\right)=\left(\sin 3x\right)\left(x^{\frac{1}{4}}-1\right)$
   35. 
       $f\left(u\right)=\sin u,g\left(x\right)=x^2+5x$
   36. 
       $f\left(u\right)=e^u,g\left(x\right)=3x^2+8$
   37. 
       $f\left(u\right)=\frac{1}{\log u},g\left(x\right)=x^4+1$
   38. 
       $f\left(u\right)=\frac{1}{\log u},g\left(x\right)=x^{\frac{1}{2}}+2x$
   39. 
       $f\left(u\right)=u,g\left(x\right)=\frac{2x}{e^x}$
2. 1. 
      $8{\left(x+1\right)}^7$
   2. 
      $\frac{1}{2}{\left(2x-5\right)}^{-\frac{1}{2}}·2$
   3. 
      $3{\left(\sin x\right)}^2\cos x$
   4. 
      $5{\left(\log x\right)}^4\frac{1}{x}$
   5. 
      $\cos 2x·2$
   6. 
      $\frac{1}{x^2+1}·2x$
   7. 
      $e^{\cos x}\left(-\sin x\right)$
   8. 
      $\frac{1}{e^x+\sin x}\left(e^x+\cos x\right)$
   9. 
      $\cos \left(\log x+\frac{1}{x}\right)·\left(\frac{1}{x}-\frac{1}{x^2}\right)$
   10. 
       $\frac{\sin 2x-\left(x+1\right)\cos \left(2x\right)2}{{\sin }^{2}2x}$
   11. 
       $3{\left(2x^2+3\right)}^2·4x$
   12. 
       $-\sin \left(\sin 5x\right)·\cos 5x·5$
   13. 
       $\frac{1}{\cos 2x}\left(-\sin 2x·2\right)$
   14. 
       $\cos {\left(2x+5\right)}^2·2\left(2x+5\right)·2$
   15. 
       $\cos \left(\cos \left(x+1\right)\right)·\left(-\sin \left(x+1\right)\right)$
   16. 
       $\cos e^x·e^x$
   17. 
       $-\frac{4\left(3x-1\right)·3}{{\left(3x-1\right)}^8}$
   18. 
       $\frac{-3{\left(4x\right)}^{2}4}{{\left(4x\right)}^6}$
   19. 
       $\frac{-2\sin 2x·2}{{\left(\sin 2x\right)}^4}$
   20. 
       $\frac{-2\cos 2x·2}{{\left(\cos 2x\right)}^4}$
   21. 
       $\frac{-\cos 3x·3}{{\sin }^{2}3x}$
   22. 
       ${\cos }^{2}x-{\sin }^{2}x$
   23. 
       $2x{e}^x+\left(x^2+1\right)e^x$
   24. 
       $\left(3x^2+2\right)\sin 3x+\left(x^3+2x\right)\cos 3x·3$
   25. 
       $\frac{-\left(\cos x-\sin x\right)}{\sin x+\cos x}$
   26. 
       $\frac{\cos 2x·2e^x-\sin 2x·e^x}{e^{2x}}$
   27. 
       $\frac{\frac{1}{x}\left(x^2+3\right)-\log x·2x}{{\left(x^2+3\right)}^2}$
   28. 
       $\frac{\cos 2x-\left(x+1\right)\left(-\sin 2x\right)2}{{\cos }^{2}2x}$
   29. 
       $2\left(e^x+x\right)+\left(2x-3\right)\left(e^x+1\right)$
   30. 
       $3x^2\left(e^{3x}+5x\right)+\left(x^3-1\right)\left(e^{3x}·3+5\right)$
   31. 
       $\frac{3x^2\left(x-1\right)-\left(x^3+1\right)}{{\left(x-1\right)}^2}$
   32. 
       $\frac{2x\left(2x+3\right)-\left(x^2-1\right)2}{{\left(2x+3\right)}^2}$
   33. 
       $\left(\frac{4}{3}{x}^{\frac{1}{3}}-e^x\right)\left(2x+1\right)+\left(x^{\frac{4}{3}}-e^x\right)2$
   34. 
       $\cos 3x·3\left(x^{\frac{1}{4}}-1\right)+\sin 3x·\frac{1}{4}{x}^{-\frac{3}{4}}$
   35. 
       $\cos \left(x^2+5x\right)·\left(2x+5\right)$
   36. 
       $e^{3x^2+8}·6x$
   37. 
       $\frac{-\frac{1}{x^4+1}·4x^3}{{\left(\log \left(x^4+1\right)\right)}^2}$
   38. 
       $\frac{-\frac{1}{x^{\frac{1}{2}}+2x}·\left(\frac{1}{2}{x}^{-\frac{1}{2}}+2\right)}{{\left(\log \left(x^{\frac{1}{2}}+2x\right)\right)}^2}$
   39. 
       $\frac{2e^x-2x{e}^x}{e^{2x}}$