Miscellaneous Exercises

Differentiate the following functions:

1. $\arcsin \sqrt{1 - 4x^2}$                                                 Ans: $\frac{-2}{\sqrt{1 - 4x^2}}$

2. $xe^{x^2}$                                                Ans: $e^{x^2}(2x^2 + 1)$

3. $\log \sin \frac{v}{2}$                                                Ans: $\frac{1}{2} \cot \frac{v}{2}$

4. $\arccos \frac{a}{y}$                                                Ans: $\frac{a}{y\sqrt{y^2 - a^2}}$

5. $\frac{x}{\sqrt{a^2 - x^2}}$                                                Ans: $\frac{a^2}{(a^2 - x^2)^{\frac{3}{2}}}$

6. $\frac{x}{1 + \log\, x}$                                                Ans: $\frac{\log\, x}{(1 + \log\, x)^2}$

7. $\log\sec(1 - 2x)$                                                Ans: $- 2\tan(1 - 2x)$

8. $x^2e^{2 - 3x}$                                                Ans: $xe^{2 - 3x}(2 - 3x)$

9. $\log \sqrt{\frac{1 - \cos t}{1 + \cos t}}$                                Ans: $\csc\, t$

   Here's how Sage tackles this one:

   [fontsize=\tiny,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: t = var("t")
   sage: diff(log(sqrt((1-cos(t))/(1+cos(t)))),t)
   (cos(t) + 1)*(sin(t)/(cos(t) + 1) + (1 - cos(t))*sin(t)/(cos(t) + 1)^2)/(2*(1 - cos(t)))
   sage: diff(log(sqrt((1-cos(t))/(1+cos(t)))),t).simplify_trig()
   -sin(t)/(cos(t)^2 - 1)
   

   Since $\cos(t)^2-1=-\sin(t)^2$, the result returned by Sage agrees with the answer given.

10. $\arcsin \sqrt{\frac{1}{2}(1 - \cos x)}$                                 Ans: $\frac{1}{2}$, for $x > 0$; $-\frac{1}{2}$, for $x < 0$.

    Here's how Sage tackles this one:

    [fontsize=\tiny,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: diff(arcsin(sqrt((1-cos(x))/2)),x)
    sin(x)/(2*sqrt(2)*sqrt(1 - (1 - cos(x))/2)*sqrt(1 - cos(x)))
    sage: diff(arcsin(sqrt((1-cos(x))/2)),x).simplify_trig()
    sin(x)/(2*sqrt(1 - cos(x))*sqrt(cos(x) + 1))
    sage: diff(arcsin(sqrt((1-cos(x))/2)),x).simplify_radical()
    sin(x)/(2*sqrt(1 - cos(x))*sqrt(cos(x) + 1))
    

    Here we see again that Sage does not simplify the result down to the final answer. Nonetheless, simplify_trig is useful. Since
    $$\sqrt(1 - \cos(x))\sqrt(\cos(x) + 1)=\sqrt(1 - \cos(x)^2)=\sqrt(\sin(x)^2) =\pm \sin(x),$$
    we see the answer given is correct (at least for the interval $-\pi<x<\pi$).

11. $\arctan \frac{2s}{\sqrt{s^2 - 1}}$                                                Ans: $\frac{2}{(1 - 5s^2)\sqrt{s^2 - 1}}$

12. $(2x - 1)\sqrt[3]{\frac{2}{1 + x}}$                                                Ans: $\frac{7 + 4x}{3(1 + x)}\sqrt[3]{\frac{2}{1 + x}}$

13. $\frac{x^3 \arcsin x}{3} + \frac{(x^2 + 2)\sqrt{1 - x^2}}{9}$                                                Ans: $x^2\arcsin\, x$

14. $\tan^2 \frac{\theta}{3} + \log \sec^2 \frac{\theta}{3}$

15. $\arctan \frac{1}{2}(e^{2x} + e^{-2x})$

16. $\left ( \frac{3}{x} \right )^{2x}$

17. $x^{\tan\, x}$

18. $\frac{(x + 2)^{\frac{1}{3}} (x^2 - 1)^{\frac{2}{5}}}{x^{\frac{3}{2}}}$

19. $e^{\sec(1 - 3x)}$

20. $\arctan \sqrt{1 - x^2}$

21. $\frac{z^2}{\cos z}$

22. $e^{\tan x^2}$

23. $\log \sin^2 \frac{1}{2} \theta$

24. $e^{ax}\log\sin\, ax$

    Here's how Sage tackles this Exercise:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: a = var("a")
    sage: diff(exp(a*x)*log(sin(a*x)),x)
    a*e^(a*x)*log(sin(a*x)) + a*e^(a*x)*cos(a*x)/sin(a*x)
    

25. $\sin\, 3\phi \cos\, \phi$

26. $\frac{a}{2\sqrt{(b - cx^n)^m}}$

27. $\frac{m + x}{1 + m^2} \cdot \frac{e^{m \arctan x}}{\sqrt{1 + x^2}}$

28. $\tan^2x - \log\sec^2x$

29. $\frac{3 \log (2 \cos x + 3 \sin x) + 2x}{13}$

30. ${\rm arccot}\frac{a}{x} + \log \sqrt{\frac{x - a}{x + a}}$

31. $(\log\tan(3 - x^2)^3$

32. $\frac{2 - 3t^{\frac{1}{2}} + 4t^{\frac{1}{3}} + t^2}{t}$

33. $\frac{(1 + x)(1 - 2x)(2 + x)}{(3 + x)(2 - 3x)}$

34. $\arctan(\log\, 3x)$

    Here's how Sage tackles this one:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: diff(arctan(log(3*x)),x)
    1/(x*(log(3*x)^2 + 1))
    

35. $\sqrt[3]{(b - ax^m)^n}$

    Here's how Sage tackles this one:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: a,b,m,n = var("a,b,m,n")
    sage: diff((b-a*x^m)^(n/3),x)
    -a*m*n*x^(m - 1)*(b - a*x^m)^(n/3 - 1)/3
    

36. $\log \sqrt{(a^2 - bx^2)^m}$

37. $\log \sqrt{\frac{y^2 + 1}{y^2 - 1}}$

38. $e^{{\rm arcsec}\, 2\theta}$

39. $\sqrt{\frac{(2 - 3x)^3}{1 + 4x}}$

40. $\frac{\sqrt[3]{a^2 - x^2}}{\cos x}$

41. $e^x\log\sin\, x$

42. $\arcsin \frac{x}{\sqrt{1 + x^2}}$

43. $\arctan\, a^x$

44. $a^{\sin^2 mx}$

    Here's how Sage tackles this one:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: a,m = var("a,m")
    sage: diff(a^(sin(m*x)^2),x)
    2*a^sin(m*x)^2*log(a)*m*cos(m*x)*sin(m*x)
    

45. $\cot^3 (\log\, ax)$

46. $(1 - 3x^2)e^{\frac{1}{x}}$

47. $\log \frac{\sqrt{1 - x^2}}{\sqrt[3]{1 + x^3}}$