Examples

^5.1

Differentiate the following^5.2:

1. $y = x^3$.

   Solution. $\frac{dy}{dx} = \frac{d}{dx}(x^3) = 3x^2$. (By VIa, n = 3.)

2. $y = ax^4 - bx^2$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} & = \frac{d}{dx} (ax^4 - bx^... ...m by\ IV}\\ & = 4ax^3 - 2bx \ \ \ {\rm by\ VIa}. \end{array}$$

3. $y = x^{\frac{4}{3}} + 5$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} & = \frac{d}{dx}(x^{\frac{4}... ...c{4}{3} x^{\frac{1}{3}} \ \ \ {\rm by\ VIa\ and\ I} \end{array}$$

4. $y = \frac{3x^3}{\sqrt[5]{x^2}} - \frac{7x}{\sqrt[3]{x^4}} + 8\sqrt[7]{x^3}$

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} & = \frac{d}{dx} \left ( 3 x... ...{7} x^{-\frac{4}{7}} \ \ \ {\rm by\ IV\ and\ VIa}. \end{array}$$

5. $y = (x^2 - 3)^5$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} &=5 (x^2 - 3)^4 \frac{d}{dx}... ...= 5}\\ 5(x^2 - 3)^4 \cdot 2x & = 10x(x^2 - 3)^4. \end{array}$$
   We might have expanded this function by the Binomial Theorem and then applied III, etc., but the above process is to be preferred.

6. $y = \sqrt{a^2 - x^2}$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} &= \frac{d}{dx}(a^2 - x^2)^{... ...-\frac{1}{2}} (-2x) = -\frac{x}{\sqrt{a^2 - x^2}}. \end{array}$$

7. $y = (3x^2 + 2)\sqrt{1 + 5x^2}$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} & = (3x^2 + 2) \frac{d}{dx} ... ...5x^2} \\ & = \frac{45x^3 + 16x}{\sqrt{1 + 5x^2}}. \end{array}$$

8. $y = \frac{a^2 + x^2}{\sqrt{a^2 - x^2}}$.

   Solution.

   $$\begin{array}{ll} \frac{dy}{dx} & = \frac{(a^2 - x^2)^{\frac... ...rac{\frac3 x^2x - x^3}{(a^2 - x^2)^{\frac{3}{2}}}. \end{array}$$

9. $5x^4 + 3x^2 - 6$.         (Ans. $\frac{dy}{dx} = 20x^3 + 6x$)

10. $y = 3cx^2 - 8dx + 5e$.         (Ans. $\frac{dy}{dx} = 6cx - 8d$)

11. $y = x^{a + b}$.         (Ans. $\frac{dy}{dx} = (a + b)x^{a + b - 1}$)

12. $y = x^n + nx + n$.         (Ans. $\frac{dy}{dx} = nx^{n - 1} + n$)

13. $f(x) = \frac{2}{3} x^3 - \frac{3}{2} x^2 + 5$.         (Ans. $f'(x) = 2x^2 - 3x$)

14. $f(x) = (a + b)x^2 + cx + d$.         (Ans. $f'(x) = 2(a + b)x + c$)

15. $\frac{d}{dx}(a + bx + cx^2) = b + 2cx$.

16. $\frac{d}{dy}(5y^m - 3y + 6) = 5my^{m - 1} - 3$.

17. $\frac{d}{dx}(2 x^{-2} + 3x^{-3}) = -4x^{-3} - 9x^{-4}$.

18. $\frac{d}{ds}(3s^{-4} - s) = -12s^{-5} - 1$.

19. $\frac{d}{dx}(4x^{\frac{1}{2}} + x^2) = 2x^{-\frac{1}{2}} + 2x$.

20. $\frac{d}{dy}(y^{-2} - 4y^{-\frac{1}{2}}) = -2y^{-3} + 2y^{-\frac{3}{2}}$.

21. $\frac{d}{dx}(2x^3 + 5) = 6x^2$.

22. $\frac{d}{dt}(3t^5 - 2t^2) = 15t^4 - 4t$.

23. $\frac{d}{d\theta}(a\theta^4 + b\theta) = 4a\theta^3 + b$.

24. $\frac{d}{d\alpha}(5 - 2\alpha^{\frac{3}{2}}) = -3\alpha^{\frac{1}{2}}$.

25. $\frac{d}{dt}(9t^{\frac{5}{3}} + t^{-1}) = 15t^{\frac{2}{3}} - t^{-2}$.

26. $\frac{d}{dx}(2x^{12} - x^9) = 24x^{11} - 9x^8$.

27. $r = c\theta^3 + d\theta^2 + e\theta$.         (Ans. $r' = 3c\theta^2 + 2d\theta + e$)

28. $y = 6x^{\frac{7}{2}} + 4x^{\frac{5}{2}} + 2x^{\frac{3}{2}}$.         (Ans. $y' = 21x^{\frac{5}{2}} + 10x^{\frac{3}{2}} + 3x^{\frac{1}{2}}$)

29. $y = \sqrt{3x} + \sqrt{3}{x} + \frac{1}{x}$.         (Ans. $y' = \frac{3}{2\sqrt{3x}} + \frac{1}{3\sqrt[3]{x^2}} - \frac{1}{x^2}$)

30. $y = \frac{a + bx + cx^2}{x}$.         (Ans. $y' = c - \frac{a}{x^2}$)

31. $y = \frac{(x - 1)^3}{x^{\frac{1}{3}}}$.         (Ans. $y' = \frac{8}{3}x^{\frac{5}{3}} - 5x^{\frac{2}{3}} + 2x^{-\frac{1}{3}} + \frac{1}{3}x^{-\frac{4}{3}}$)

32. $y = (2x3 + x^2 - 5)^3$.         (Ans. $y' = 6x(3x + 1)(2x^3 + x^2 - 5)^2$)

33. $y = (2x^3 + x^2 - 5)^3$.         (Ans. $y' = 6x(3x + 1)(2x^3 + x^2 - 5)^2$)

34. $f(x) = (a + bx^2)^{\frac{5}{4}}$.         (Ans. $f'(x) = \frac{5bx}{2}(a + bx^2)^{\frac{1}{4}}$)

35. $f(x) = (1 + 4x^3)(1 + 2x^2)$.         (Ans. $f'(x) = 4x(1 + 3x + 10x^3)$)

36. $f(x) = (a + x)\sqrt{a - x}$.         (Ans. $f'(x) = \frac{a - 3x}{2\sqrt{a - x}}$)

37. $f(x) = (a + x)^m(b + x)^n$.         (Ans. $f'(x) = (a + x)^m(b + x)^n \left [ \frac{m}{a + x} + \frac{n}{b + x} \right ]$)

38. $y = \frac{1}{x^n}$ .         (Ans. $\frac{y}{x} = -\frac{n}{x^{n + 1}}$)

39. $y = x(a^2 + x^2)\sqrt{a^2 - x^2}$.         (Ans. $\frac{dy}{dx} = \frac{a^4 + a^2x^2 - 4x^4}{\sqrt{a^2 - x^2}}$)

40. Differentiate the following functions:
    $$\begin{array}{lll} (a)\ \ \frac{d}{dx}(2x^3 - 4x + 6) & (e)\... ...\ \frac{d}{dx}(2x^{\frac{1}{3}} + 2x^{\frac{5}{3}}) \end{array}$$

41. $y = \frac{2x^4}{b^2 - x^2}$.         (Ans. $\frac{dy}{dx} = \frac{8b^2x^3 - 4x^5}{(b^2 - x^2)^2}$)

42. $y = \frac{a - x}{a + x}$.         (Ans. $\frac{dy}{dx} = -\frac{2a}{(a + x)^2}$)

43. $s = \frac{t^3}{(1 + t)^2}$.         (Ans. $\frac{ds}{dt} = \frac{3t^2 + t^3}{(1 + t)^3}$)

44. $f(s) = \frac{(s + 4)^2}{s + 3}$.         (Ans. $f'(s) = \frac{(s + 2)(s + 4)}{(s + 3)^2}$)

45. $f(\theta) = \frac{\theta}{\sqrt{a - b\theta^2}}$.         (Ans. $f'(\theta) = \frac{a}{(a - b\theta^2)^{\frac{3}{2}}}$)

46. $F(r) = \sqrt{\frac{1 + r}{1 - r}}$.         (Ans. $F'(r) = \sqrt{1}{(1 - r)\sqrt{1 - r^2}}$)

47. $\psi(y) = \left ( \frac{y}{1 - y} \right )^m$.         (Ans. $\psi'(y) = \frac{my^{m - 1}}{(1 - y)^{m + 1}}$)

48. $\phi(x) = \frac{2x^2 - 1}{x\sqrt{1 + x^2}}$.         (Ans. $\phi'(x) = \frac{1 + 4x^2}{x^2(1 + x^2)^{\frac{3}{2}}}$)

49. $y = \sqrt{2px}$.         (Ans. $y' = \frac{p}{y}$)

50. $y = \frac{b}{a}\sqrt{a^2 - x^2}$.         (Ans. $y' = -\frac{b^2x}{a^2y}$)

51. $y = (a^{\frac{2}{3}} - x^{\frac{2}{3}})^{\frac{3}{2}}$.         (Ans. $y' = -\sqrt[3]{\frac{y}{x}}$)

52. $r = \sqrt{a\phi} + c\sqrt{\phi^3}$.         (Ans. $r' = \frac{\sqrt{a} + 3c\phi}{2\sqrt{\phi}}$)

53. $u = \frac{v^c + v^d}{cd}$.         (Ans. $u' = \frac{v^{c - 1}}{d} + \frac{v^{d - 1}}{c}$)

54. $p = \frac{(q + 1)^{\frac{3}{2}}}{\sqrt{q - 1}}$.         (Ans. $p' = \frac{(q - 2)\sqrt{q + 1}}{(q - 1)^{\frac{3}{2}}}$)

55. Differentiate the following functions:
    $$\begin{array}{lll} (a) \ \ \frac{d}{dx} \left ( \frac{a^2 - ... ...(i) \ \ \frac{d}{dt} \sqrt{\frac{1 + t^2}{1 - t^2}} \end{array}$$