Examples

Differentiate the following^5.7:

1. $y = \log(x + a)$                                                Ans: $\frac{dy}{dx} = \frac{1}{x + a}$

2. $y = \log(ax + b)$                                                Ans: $\frac{dy}{dx} = \frac{a}{ax + b}$

3. $y = \log \frac{1 + x^2}{1 - x^2}$                                                Ans: $\frac{dy}{dx} = \frac{4x}{1 - x^4}$

4. $y = \log(x^2 + x)$                                                Ans: $y' = \frac{2x + 1}{x^2 + x}$

5. $y = \log(x^3 - 2x + 5)$                                                Ans: $y' = \frac{3x^2 - 2}{x^3 - 2x + 5}$

6. $y = \log_a(2x + x^3)$                                                Ans: $y' = \log_a e \cdot \frac{2 + 3x^2}{2x + x^3}$

7. $y = x\log x$                                                Ans: $y' = \log x + 1$

8. $f(x) = \log (x^3)$                                                Ans: $f'(x) = \frac{3}{x}$

9. $f(x) = \log^3 x$                                                Ans: $f'(x) = \frac{3 \log^2 x}{x}$

   (Hint: $\log^3x = (\log x)^3$. Use first VI, $v = \log x$, $n = 3$; and then VIIIa.)

10. $f(x) = \log \frac{a + x}{a - x}$                                                Ans: $f'(x) = \frac{2a}{a^2 - x^2}$

11. $f(x) = \log (x + \sqrt{1 + x^2})$                                                Ans: $f'(x) = \frac{1}{\sqrt{1 + x^2}}$

12. $\frac{d}{dx} e^{ax} = ae^{ax}$

13. $\frac{d}{dx} e^{4x + 5} = 4e^{4x + 5}$

14. $\frac{d}{dx} a^{3x} = 3a^{3x} \log a$

15. $\frac{d}{dt} \log(3 - 2t^2) = \frac{4t}{2t^2 - 3}$

16. $\frac{d}{dy} \log \frac{1 + y}{1 - y} = \frac{2}{1 - y^2}$

17. $\frac{d}{dx}e^{b^2 + x^2} = 2xe^{b^2 + x^2}$

18. $\frac{d}{d\theta} a^{\log a} = \frac{1}{\theta} a^{\log \theta} \log a$

19. $\frac{d}{ds}b^{s^2} = 2x \log b \cdot b^{s^2}$

20. $\frac{d}{dv} ae^{\sqrt{v}} = \frac{ae^{\sqrt{v}}}{2\sqrt{v}}$

21. $\frac{d}{dx} a^{e^x} = \log a \cdot a^{e^x} \cdot e^x$

22. $y = 7^{x^2 + 2x}$                                                Ans: $y' = 2\log 7 \cdot (x + 1) 7^{x^2 + 2x}$

23. $y = c^{a^2 - x^2}$                                                Ans: $y' = -2x \log c \cdot c^{a^2 - x^2}$

24. $y = \log \frac{e^x}{1 + e^x}$                                                Ans: $\frac{dy}{dx} = \frac{1}{1 + e^x}$

25. $\frac{d}{dx} \left [ e^x ( 1- x^2 \right ] = e^x (1 - 2x - x^2)$

26. $\frac{d}{dx} \left ( \frac{e^x - 1}{e^x + 1} \right ) = \frac{2e^x}{(e^x + 1)^2}$

27. $\frac{d}{dx} \left ( x^2 e^{ax} \right ) = xe^{ax}(ax + 2)$

28. $y = \frac{a}{2} (e^{\frac{x}{a}} - e^{-\frac{x}{a}})$                                                Ans: $\frac{dy}{dx} = \frac{1}{2} (e^{\frac{x}{a}} + e^{-\frac{x}{a}})$

29. $y = \frac{e^x - e^{-x}}{e^x + e^{-x}}$                                                Ans: $\frac{dy}{dx} = \frac{4}{(e^x + e^{-x}))^2}$

30. $y = x^na^x$                                                Ans: $y' = a^xx^{n - 1}(n + x\log a)$

31. $y = x^x$                                                Ans: $y' = x^x(\log x + 1)$
32. $y = x^{\frac{1}{x}}$                                                 Ans: $y' = \frac{x^{\frac{1}{x}} (1 - \log x)}{x^2}$

33. $y = x^{\log x}$                                                Ans: $y' = \log (x^2) \cdot x^{\log x - 1}$

34. $f(y) = \log y \cdot e^y$                                                Ans: $f'(y) = e^y \left ( \log y + \frac{1}{y} \right )$

35. $f(s) = \frac{\log s}{e^s}$                                                Ans: $f'(s) = \frac{1 - s \log s}{s e^s}$

36. $f(x) = \log(\log x)$                                                Ans: $f'(x) = \frac{1}{x \log x}$

37. $F(x) = \log^4(\log x)$                                                Ans: $F'(x) = \frac{4 \log^3 (\log x)}{x \log x}$

38. $\phi (x) = \log(\log^4x)$                                                Ans: $\phi'(x) = \frac{4}{x \log x}$

39. $\psi(y) = \log \sqrt{\frac{1 + y}{1 - y}}$                                                Ans: $\psi'(y) = \frac{1}{1 - y^2}$

40. $f(x) = \log \frac{\sqrt{x^2 + 1} - x}{\sqrt{x^1 + 1} + x}$                                                Ans: $f'(x) = -\frac{2}{\sqrt{1 + x^2}}$

41. $y = x^{\frac{1}{\log x}}$                                                Ans: $\frac{dy}{dx} = 0$

42. $y = e^{x^x}$                                                Ans: $\frac{dy}{dx} = e^{x^x}(1 + \log x)x^x$

43. $y = \frac{c^x}{x^x}$                                                Ans: $\frac{dy}{dx} = \left ( \frac{c}{x} \right )^x \left ( \log \frac{c}{x} - 1 \right )$

44. $y = \left ( \frac{x}{n} \right )^{nx}$                                                Ans: $\frac{dy}{dx} = n \left ( \frac{x}{n} \right )^{nx} \left ( 1 + \log \frac{x}{n} \right )$

45. $w = v^{e^v}$                                                Ans: $\frac{dw}{dv} = v^{e^v} e^v \left ( \frac{1 + v \log v}{v} \right )$

46. $z = \left ( \frac{a}{t} \right )^t$                                                Ans: $\frac{dz}{dt} = \left ( \frac{a}{t} \right )^t (\log a - \log t - 1)$

47. $y = x^{x^n}$                                                Ans: $\frac{dy}{dx} = x^{x^n + n - 1}(n \log x + 1)$

48. $y = x^{x^x}$                                                Ans: $\frac{dy}{dx} = x^{x^x} x^x \left ( \log x + \log^2 x + \frac{1}{x} \right )$

49. $y = a^{\frac{1}{\sqrt{a^2 - x^2}}}$                                                Ans: $\frac{dy}{dx} = \frac{xy \log a}{(a^2 - x^2)^{\frac{3}{2}}}$

50. Compute the following derivatives:
    $$\begin{array}{lll} (a)\ \ \frac{d}{dx} x^2 \log x & (f)\ \ \... ...1}{x} \right )^x & (o)\ \ \frac{d}{dx} (x^2 + 4)^x. \end{array}$$

51. $y = \frac{(x + 1)^2}{(x + 2)^3 (x + 3)^4}$                                                 Ans: $\frac{dy}{dx} = -\frac{(x + 1)(5x^2 + 14x + 5)}{(x + 2)^4 (x + 3)^5}$

52. $y = \frac{((x - 1)^{\frac{5}{2}}}{(x - 2)^{\frac{3}{4}}(x - 3)^{\frac{7}{3}}}$                                                Ans: $\frac{dy}{dx} = -\frac{(x - 1)^{\frac{3}{2}}(7x^2 + 30x - 97)}{12(x - 2)^{\frac{7}{4}}(x - 3)^{\frac{10}{3}}}$

53. $y = x \sqrt{1 - x} (1 + x)$                                                 Ans: $\frac{dy}{dx} = \frac{2 + x - 5x^2}{2\sqrt{1 - x}}$

54. $y = \frac{x(1 + x^2)}{\sqrt{1 - x^2}}$                                                 Ans: $\frac{dy}{dx} = \frac{1 + 3x^2 - 2x^4}{(1 - x^2}^{\frac{3}{2}}$

55. $y = x^5(a + 3x)^3(a - 2x)^2$                         Ans: $\frac{dy}{dx} = 5x^4(a + 3x)^2(a - 2x)(a^2 + 2ax - 12x^2)$