## 2017年2月7日火曜日

### 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)

1-40.

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. $f\left(u\right)={u}^{3},g\left(x\right)=\mathrm{sin}x$

4. $f\left(u\right)={u}^{5},g\left(x\right)=\mathrm{log}x$

5. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)=2x$

6. $f\left(u\right)=\mathrm{log}u,g\left(x\right)={x}^{2}+1$

7. $f\left(u\right)={e}^{u},g\left(x\right)=\mathrm{cos}x$

8. $f\left(u\right)=\mathrm{log}u,g\left(x\right)={e}^{x}+\mathrm{sin}x$

9. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)=\mathrm{log}x+\frac{1}{x}$

10. $f\left(u\right)=\frac{\frac{1}{2}u+1}{\mathrm{sin}u},g\left(x\right)=2x$

11. $f\left(u\right)={u}^{3},g\left(x\right)=2{x}^{2}+3$

12. $f\left(u\right)=\mathrm{cos}u,g\left(x\right)=\mathrm{sin}5x$

13. $f\left(u\right)=\mathrm{log}u,g\left(x\right)=\mathrm{cos}2x$

14. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)={\left(2x+5\right)}^{2}$

15. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)=\mathrm{cos}\left(x+1\right)$

16. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)={e}^{x}$

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. $f\left(u\right)=\frac{1}{{u}^{2}},g\left(x\right)=\mathrm{sin}2x$

20. $f\left(u\right)=\frac{1}{{u}^{2}},g\left(x\right)=\mathrm{cos}2x$

21. $f\left(u\right)=\frac{1}{\mathrm{sin}u},g\left(x\right)=3x$

22. $f\left(u\right)=\frac{1}{\frac{1}{2}u},g\left(x\right)=2\mathrm{sin}x\mathrm{cos}x=\mathrm{sin}2x$

23. $f\left(u\right)=\left({u}^{2}+1\right){e}^{u},g\left(x\right)=x$

24. $f\left(u\right)=\left({u}^{3}+2u\right)\left(\mathrm{sin}3u\right),g\left(x\right)=x$

25. $f\left(u\right)=\frac{1}{u},g\left(x\right)=\mathrm{sin}x+\mathrm{cos}x$

26. $f\left(u\right)=\frac{\mathrm{sin}2u}{{e}^{u}},g\left(x\right)=x$

27. $f\left(u\right)=\frac{\mathrm{log}u}{{u}^{2}+3},g\left(x\right)=x$

28. $f\left(u\right)=\frac{u+1}{\mathrm{cos}2u},g\left(x\right)=x$

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. $f\left(u\right)=u,g\left(x\right)=\left(\mathrm{sin}3x\right)\left({x}^{\frac{1}{4}}-1\right)$

35. $f\left(u\right)=\mathrm{sin}u,g\left(x\right)={x}^{2}+5x$

36. $f\left(u\right)={e}^{u},g\left(x\right)=3{x}^{2}+8$

37. $f\left(u\right)=\frac{1}{\mathrm{log}u},g\left(x\right)={x}^{4}+1$

38. $f\left(u\right)=\frac{1}{\mathrm{log}u},g\left(x\right)={x}^{\frac{1}{2}}+2x$

39. $f\left(u\right)=u,g\left(x\right)=\frac{2x}{{e}^{x}}$

1. $8{\left(x+1\right)}^{7}$

2. $\frac{1}{2}{\left(2x-5\right)}^{-\frac{1}{2}}·2$

3. $3{\left(\mathrm{sin}x\right)}^{2}\mathrm{cos}x$

4. $5{\left(\mathrm{log}x\right)}^{4}\frac{1}{x}$

5. $\mathrm{cos}2x·2$

6. $\frac{1}{{x}^{2}+1}·2x$

7. ${e}^{\mathrm{cos}x}\left(-\mathrm{sin}x\right)$

8. $\frac{1}{{e}^{x}+\mathrm{sin}x}\left({e}^{x}+\mathrm{cos}x\right)$

9. $\mathrm{cos}\left(\mathrm{log}x+\frac{1}{x}\right)·\left(\frac{1}{x}-\frac{1}{{x}^{2}}\right)$

10. $\frac{\mathrm{sin}2x-\left(x+1\right)\mathrm{cos}\left(2x\right)2}{{\mathrm{sin}}^{2}2x}$

11. $3{\left(2{x}^{2}+3\right)}^{2}·4x$

12. $-\mathrm{sin}\left(\mathrm{sin}5x\right)·\mathrm{cos}5x·5$

13. $\frac{1}{\mathrm{cos}2x}\left(-\mathrm{sin}2x·2\right)$

14. $\mathrm{cos}{\left(2x+5\right)}^{2}·2\left(2x+5\right)·2$

15. $\mathrm{cos}\left(\mathrm{cos}\left(x+1\right)\right)·\left(-\mathrm{sin}\left(x+1\right)\right)$

16. $\mathrm{cos}{e}^{x}·{e}^{x}$

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. $\frac{-2\mathrm{sin}2x·2}{{\left(\mathrm{sin}2x\right)}^{4}}$

20. $\frac{-2\mathrm{cos}2x·2}{{\left(\mathrm{cos}2x\right)}^{4}}$

21. $\frac{-\mathrm{cos}3x·3}{{\mathrm{sin}}^{2}3x}$

22. ${\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x$

23. $2x{e}^{x}+\left({x}^{2}+1\right){e}^{x}$

24. $\left(3{x}^{2}+2\right)\mathrm{sin}3x+\left({x}^{3}+2x\right)\mathrm{cos}3x·3$

25. $\frac{-\left(\mathrm{cos}x-\mathrm{sin}x\right)}{\mathrm{sin}x+\mathrm{cos}x}$

26. $\frac{\mathrm{cos}2x·2{e}^{x}-\mathrm{sin}2x·{e}^{x}}{{e}^{2x}}$

27. $\frac{\frac{1}{x}\left({x}^{2}+3\right)-\mathrm{log}x·2x}{{\left({x}^{2}+3\right)}^{2}}$

28. $\frac{\mathrm{cos}2x-\left(x+1\right)\left(-\mathrm{sin}2x\right)2}{{\mathrm{cos}}^{2}2x}$

29. $2\left({e}^{x}+x\right)+\left(2x-3\right)\left({e}^{x}+1\right)$

30. $3{x}^{2}\left({e}^{3x}+5x\right)+\left({x}^{3}-1\right)\left({e}^{3x}·3+5\right)$

31. $\frac{3{x}^{2}\left(x-1\right)-\left({x}^{3}+1\right)}{{\left(x-1\right)}^{2}}$

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. $\mathrm{cos}3x·3\left({x}^{\frac{1}{4}}-1\right)+\mathrm{sin}3x·\frac{1}{4}{x}^{-\frac{3}{4}}$

35. $\mathrm{cos}\left({x}^{2}+5x\right)·\left(2x+5\right)$

36. ${e}^{3{x}^{2}+8}·6x$

37. $\frac{-\frac{1}{{x}^{4}+1}·4{x}^{3}}{{\left(\mathrm{log}\left({x}^{4}+1\right)\right)}^{2}}$

38. $\frac{-\frac{1}{{x}^{\frac{1}{2}}+2x}·\left(\frac{1}{2}{x}^{-\frac{1}{2}}+2\right)}{{\left(\mathrm{log}\left({x}^{\frac{1}{2}}+2x\right)\right)}^{2}}$

39. $\frac{2{e}^{x}-2x{e}^{x}}{{e}^{2x}}$