1. \[ \frac{dy}{dx} = 3(x^2 – 3x + 5)^2(2x – 3) \]
2. \[ f'(x) = -32(15 – 8x)^3 \]
3. \[ g'(t) = -18t^2(2t^3 – 1)^{-4} \]
4. \[ \frac{dz}{dx} = -8(5x^5 – x^4)^{-9}(25x^4 – 4x^3) \]
5. \( \frac{dy}{dx} = -32x(3x^2 – 8)^3(-4x^2 + 1)^3\) \(+ 18x(3x^2 – 8)^2(-4x^2 + 1)^4 \)
6. \[ f'(u) = \frac{2u(u^3 – 3u – 1)}{(u^2 – 1)^2} \]
7. \[ \frac{dy}{dx} = \frac{8(x-1)}{(x+3)^3} \]
8. \[ g'(t) = \frac{-12t(3t^2 + 2)(t^2 + 2t + 1)}{(2t^3 – 1)^3} \]
9. \[ y’ = \frac{-1}{\sqrt{1-2x}} \]
10. \[ u’ = \frac{1-4t-24t^2}{2\sqrt{1+t-2t^2-8t^3}} \]
11. \[ h'(x) = \frac{2x^5}{\sqrt{x^4 – 1}} + 2x\sqrt{x^4 – 1} \]
12. \[ g'(x) = \frac{1}{(x^2+1)^{\frac{3}{2}}} \]
13. \[ y’ = \frac{2\sqrt{3x^2-1}}{3\sqrt[3]{(2x+1)^2}} + \frac{3x\sqrt[3]{2x+1}}{\sqrt{3x^2-1}} \]
14. \[ z’ = -\frac{(1-3x^2)^2}{\sqrt{x}(\sqrt{x}+1)^3} – \frac{12x(1-3x^2)}{(\sqrt{x}+1)^2} \]
15. \[ h'(t) = \frac{3-t}{2(1-t)^{\frac{3}{2}}} \]
16. \[ z’ = -\frac{2t}{3(1+t^2)^{\frac{4}{3}}} \]
17. \[ z’ = \frac{ax^2}{\sqrt[3]{(b-ax^3)^2}} \]
18. \[ f'(x) = \frac{1}{\sqrt{(b^2-x^2)^3}} \]
19. \[ y’ = \frac{-1}{\sqrt{1+x}(1+\sqrt{1+x})^2} \]
20. \[ f'(x) = \frac{3x^2-2(a+b+c)x+ab+ac+bc}{2\sqrt{(x-a)(x-b)(x-c)}} \]
21. \[ y’ = \frac{1+2\sqrt{x}}{6\sqrt{x}(x+\sqrt{x})^{\frac{2}{3}}} \]
22. \[ y’ = \frac{1+2\sqrt{x}+2\sqrt{x}\sqrt{x+\sqrt{x}}}{8\sqrt{x}\sqrt{x+\sqrt{x}}\sqrt{x+\sqrt{x+\sqrt{x}}}} \]
23. \[ y’ = 4\sec^2 4x \]
24. \[ y’ = -\operatorname{cosec}^2 \frac{x}{2} \]
25. \[ u’ = -3x^2\operatorname{sen}(x^3) \]
26. \[ y’ = -3\operatorname{sen} x \cos^2 x \]
27. \[ y’ = 4x^3\sec^2(x^4) + 4\tan^3 x \sec^2 x \]
28. \[ z’ = -\frac{1}{2\sqrt{x}}\operatorname{sen}\sqrt{x} \]
29. \[ u’ = -\frac{\operatorname{sen} x}{2\sqrt{\cos x}} \]
30. \[ y’ = -\frac{\operatorname{sen}\sqrt{x}}{4\sqrt{x}\sqrt{\cos\sqrt{x}}} \]
31. \[ y’ = \frac{\sec^2 3x}{(\tan 3x)^{\frac{2}{3}}} \]
32. \[ y’ = -\frac{2x\operatorname{cosec}^2\sqrt[3]{1+x^2}}{3(1+x^2)^{\frac{2}{3}}} \]
33. \[ y’ = -\frac{2\tan x}{\sqrt{\sec x}} \]
34. \[ y’ = \frac{2}{x^3}\operatorname{cosec}\frac{1}{x^2}\cot\frac{1}{x^2} \]
35. \( y’ = \frac{-3}{\sqrt{x}(1+\sqrt{x})^2}\operatorname{sen}^2\left[\frac{1-\sqrt{x}}{1+\sqrt{x}}\right]\)\(\cos\left[\frac{1-\sqrt{x}}{1+\sqrt{x}}\right] \)
36. \[ y’ = \frac{2\sec^2 x}{(\sec^2 x + 1)^{\frac{3}{2}}} \]
37. \[ y’ = \frac{\cos x}{(1-\operatorname{sen} x)^2}\sqrt{\frac{1-\operatorname{sen} x}{1+\operatorname{sen} x}} \]
38. \[ y’ = -\frac{(1-x^2)\operatorname{cosec}^2(x+\frac{1}{x})}{2x^2\sqrt{1+\cot(x+\frac{1}{x})}} \]
39. \[ y’ = -\frac{\operatorname{cosec}^2(\frac{x}{2})}{2(1-\cot^2(\frac{x}{2}))^{\frac{3}{2}}} \]
40. \[ y’ = \frac{(a-b)\operatorname{sen} 2x}{2\sqrt{a\operatorname{sen}^2 x + b\cos^2 x}} \]
41. \[ y’ = \operatorname{sen} x \operatorname{sen}(\cos x) \]
42. \[ y’ = -2x\operatorname{sen} x^2 \cos(\cos x^2) \]
43. \[ y’ = -4\operatorname{sen} 4x \operatorname{sen}(2\cos 4x) \]
44. \[ y’ = \cos x \cos(\operatorname{sen} x) \cos(\operatorname{sen}(\operatorname{sen} x)) \]
45. \[ y’ = \operatorname{sen} x \operatorname{sen}(2\cos x) + \cos x \operatorname{sen}(2\operatorname{sen} x) \]
46. \[ y’ = \frac{\cos x}{2\sqrt{\operatorname{sen} x}}\sec^2(\sqrt{\operatorname{sen} x})\cos(\tan(\sqrt{\operatorname{sen} x}) \]
47. \[ y’ = \operatorname{sen} 2x \sec^2(\operatorname{sen}^2 x) \]
48. \[ y’ = -6x e^{-3x^2+1} \]
49. \[ y’ = \frac{\ln 2}{2\sqrt{x}} 2^{\sqrt{x}} \]
50. \[ y’ = x^{n-1} a^{-x^2} (n – 2x^2 \ln a) \]
51. \[ y’ = \frac{\ln 3}{t^2} \operatorname{cosec}^2\left(\frac{1}{t}\right) 3^{\cot\left(\frac{1}{t}\right)} \]
52. \[ y’ = (\ln 2)(\ln 3)\operatorname{sen}(2x)3^{\operatorname{sen}^2 x}2^{3^{\operatorname{sen}^2 x}} \]
53. \[ y’ = \frac{1}{2(\ln 5)x\sqrt{\log_5 x}} \]
54. \[ y’ = \frac{1}{x} – 1 \]
55. \[ y’ = \frac{1-2t\ln t}{t e^{2t}} \]
56. \[ y’ = \frac{8e^{4x}}{e^{8x}-1} \]
57. \[ y’ = e^{x\ln x}(1+\ln x) \]
58. \[ y’ = \frac{x-5}{2(x+1)(x-2)} \]
59. \[ y’ = \frac{1}{x^2-1} \]
60. \[ y’ = \frac{3}{x} + \cot x \]
61. \[ y’ = -\frac{1}{x^2} \tan \frac{x-1}{x} \]
62. \[ G'(2) = 20 \]
63. \[ F'(0) = -30 \]
64. \[ (f \circ g)'(x) = -\frac{3(3x^2+10x+3)}{2(x+1)^4} \]
65. \( h'(x) = [3u^2 – 4u] (2)\) \(= 6(2x-1)^2 – 8(2x-1)\) \(= 24x^2 – 40x + 14 \)
66. \[ h'(x) = \frac{1}{2\sqrt{v}}(6x^2) = \frac{3x^2}{\sqrt{2x^3-4}} \]
67. \[ h'(x) = 5t^4\left(-\frac{1}{\sqrt{x}}\right) = \frac{-5(1-2\sqrt{x})^4}{\sqrt{x}} \]
68. \[ h'(x) = \frac{-2bc}{(b+cx)^2} \]
69. \[ h'(x) = \left(-\frac{1}{v^2}\right)\left(\frac{-ax}{\sqrt{a^2-x^2}}\right) = \frac{x}{a(a^2-x^2)^{\frac{3}{2}}} \]
70. \( \frac{dy}{dx} = (9u^2 – 16u^3)(2x)\) \(= 18x(x^2-1)^2 – 32x(x^2-1)^3 \)
71. \[ \frac{dy}{dx} = 5v^4(2b) = 10b(3a+2bx)^4 \]
72. \[ \frac{dy}{dx} = 4t^3\left(\frac{a}{c}\right) = \frac{4a(ax+b)^3}{c^4} \]
73. \[ \frac{dy}{dx} = -\frac{1}{2v^{\frac{3}{2}}}(6x) = -\frac{3x}{(3x^2-1)^{\frac{3}{2}}} \]
74. \( 12x + y + 11 = 0\), \( x – 12y + 13 = 0 \)
75. \( y = \frac{3}{4}\), \( x = 0 \)
76. \( 7x – 18y – 13 = 0\), \( 54x + 21y – 47 = 0 \)
77. \( x – 12y – 17 = 0\), \( 12x + y + 86 = 0 \)
78. \( 8x – y – 3 = 0\), \(2x + 16y – 17 = 0 \)
79. \( y + 4x – 1 – \pi = 0\), \(16y – 4x + \pi – 16 = 0 \)
80. \( y – 12x + 17 = 0\), \(12y + x – 86 = 0 \)
81. \( 6y + 15x – 5\pi + 3\sqrt{3} = 0\), \(30y – 12x + 4\pi + 15\sqrt{3} = 0 \)
82. En (1,0): \(y – 2x + 2 = 0 \)
    En (2,0): \(y + x – 2 = 0 \)
    En (3,0): \(y – 2x + 6 = 0 \)
83. \[ (0,0), \, (4,0), \, (2,16) \]
84. \( 2y – x – 4 = 0\), \(2y – x + 4 = 0\). Son paralelas.
85. \[ y + x = 0, \quad 9y + x = 0 \]
86. \[ y – 5x + 2 = 0 \]
87. \[ 2y – x – 2\ln 2 = 0 \]

En los problemas del 1 al 61, derivar la función indicada, considerando que las expresiones \(a\), \(b\) y \(c\) son constantes.

1. \[ y = (x^2 – 3x + 5)^3 \]
2. \[ f(x) = (15 – 8x)^4 \]
3. \[ g(t) = (2t^3 – 1)^{-2} \]
4. \[ z = \frac{1}{(5x^5 – x^4)^8} \]
5. \[ y = (3x^2 – 8)^3(-4x^2 + 1)^4 \]
6. \[ f(u) = \frac{2u^3 + 1}{u^2 – 1} \]
7. \[ y = \left(\frac{x – 1}{x + 3}\right)^2 \]
8. \[ g(t) = \frac{3t^2 + 2}{2t^3 – 1} \]
9. \[ y = \sqrt{1 – 2x} \]
10. \[ u = \sqrt{1 + t – 2t^2 – 8t^3} \]
11. \[ h(x) = x^2\sqrt{x^4 – 1} \]
12. \[ g(x) = \frac{x}{\sqrt{x^2 + 1}} \]
13. \[ y = \sqrt{3x^2 – 1}\sqrt[3]{2x + 1} \]
14. \[ z = (1 – 3x^2)^2(\sqrt{x} + 1)^{-2} \]
15. \[ h(t) = \frac{1 + t}{\sqrt{1 – t}} \]
16. \[ z = \sqrt[3]{\frac{1}{1 + t^2}} \]
17. \[ z = \sqrt[3]{b + ax^3} \]
18. \[ f(x) = \frac{x}{\sqrt[3]{b^2 + x^2}} \]
19. \[ y = \frac{1 – \sqrt{1 + x}}{1 + \sqrt{1 + x}} \]
20. \[ f(x) = \sqrt{(x – a)(x – b)(x – c)} \]
21. \[ y = \sqrt[3]{x + \sqrt{x}} \]
22. \[ y = \sqrt{x + \sqrt{x + \sqrt{x}}} \]
23. \[ y = \tan 4x \]
24. \[ y = 2\cot \frac{x}{2} \]
25. \[ u = \cos(x^3) \]
26. \[ y = \cos^3 x \]
27. \[ y = \tan(x^4) + \tan^4 x \]
28. \[ z = \cos\sqrt{x} \]
29. \[ u = \sqrt{\cos x} \]
30. \[ y = \sqrt{\cos\sqrt{x}} \]
31. \[ y = \sqrt[3]{\tan 3x} \]
32. \[ y = \cot\sqrt[3]{1 + x^2} \]
33. \[ y = \frac{4}{\sqrt{\sec x}} \]
34. \[ y = \operatorname{cosec} \frac{1}{x^2} \]
35. \[ y = \operatorname{sen}^3 \left[ \frac{1 – \sqrt{x}}{1 + \sqrt{x}} \right] \]
36. \[ y = \frac{\tan x}{\sqrt{\sec^2(x) + 1}} \]
37. \[ y = \sqrt{\frac{1 + \operatorname{sen} x}{1 – \operatorname{sen} x}} \]
38. \[ y = \sqrt{1 + \cot\left(x + \frac{1}{x}\right)} \]
39. \[ y = \frac{\cot\left(\frac{x}{2}\right)}{\sqrt{1 – \cot^2\left(\frac{x}{2}\right)}} \]
40. \[ y = \sqrt{a\operatorname{sen}^2 x + b\cos^2 x} \]
41. \[ y = \cos(\cos x) \]
42. \[ y = \operatorname{sen}(\cos x^2) \]
43. \[ y = \operatorname{sen}^2(\cos 4x) \]
44. \[ y = \operatorname{sen}(\operatorname{sen}(\operatorname{sen} x)) \]
45. \[ y = \cos^2(\cos x) + \operatorname{sen}^2(\operatorname{sen} x) \]
46. \[ y = \operatorname{sen}(\tan\sqrt{\operatorname{sen} x}) \]
47. \[ y = \tan(\operatorname{sen}^2 x) \]
48. \[ y = e^{-3x^2 + 1} \]
49. \[ y = 2^{\sqrt{x}} \]
50. \[ y = x^n e^{-x^2} \]
51. \[ y = 3^{\cot\left(\frac{1}{x}\right)} \]
52. \[ y = 2^{x\operatorname{sen}^2 x} \]
53. \[ y = \sqrt{\log_5 x} \]
54. \[ y = \ln\left(\frac{x}{e^x}\right) \]
55. \[ y = \frac{\ln t}{e^{2t}} \]
56. \[ y = \ln\frac{e^{4x} – 1}{e^{4x} + 1} \]
57. \[ y = e^{x\ln x} \]
58. \[ y = \ln\left(\frac{x + 1}{\sqrt{x – 2}}\right) \]
59. \[ y = \ln\left(\frac{x – 1}{x + 1}\right)^{\frac{1}{2}} \]
60. \[ y = \ln(x^3 \operatorname{sen} x) \]
61. \[ y = \ln\left(\cos \frac{x – 1}{x}\right) \]

62. Si \( G(x) = (g(x))^3 \), \( g(2) = 125 \) y \( g'(2) = 150 \), hallar \( G'(2) \).
63. Si \( F(t) = [f(\operatorname{sen} t)]^2 \), \( f(0) = -3 \) y \( f'(0) = 5 \), hallar \( F'(0) \).
64. Dadas \( f(u) = \frac{1}{3}u^3 – 3u + 5 \) y \( g(x) = \frac{x – 1}{x + 1} \), hallar la derivada de \( f \circ g \) de dos maneras:

    1. encontrando \( (f \circ g)(x) \) y derivando este resultado.
    2. aplicando la regla de la cadena.

En los ejercicios del 65 al 69, hallar \( h'(x) \) si: \( h(x) = (f \circ g)(x) = f(g(x)) \).

65. \[ f(u) = u^3 – 2u^2 – 5, \quad g(x) = 2x – 1 \]
66. \[ f(v) = \sqrt{v}, \quad g(x) = 2x^3 – 4 \]
67. \[ f(t) = t^5, \quad g(x) = 1 – 2\sqrt{x} \]
68. \[ f(u) = \frac{b – u}{b + u}, \quad g(x) = cx \]
69. \[ f(v) = \frac{1}{v}, \quad g(x) = a\sqrt{a^2 – x^2} \]

En los ejercicios del 70 al 73, hallar \( \frac{dy}{dx} \).

70. \[ y = 3u^3 – 4u^4 – 1, \quad u = x^2 – 1 \]
71. \[ y = v^3, \quad v = 3a + 2bx \]
72. \[ y = t^4, \quad t = \frac{ax + b}{c} \]
73. \[ y = \frac{1}{\sqrt{v}}, \quad v = 3x^2 – 1 \]

En los ejercicios del 74 al 81, hallar la recta tangente y la recta normal al gráfico de la función dada en el punto \( (a, f(a)) \), para el valor indicado de \( a \).

74. \[ f(x) = (2x^2 – 1)^3, \quad a = -1 \]
75. \[ f(x) = \frac{3}{(2 – x^2)^2}, \quad a = 0 \]
76. \[ f(x) = \frac{x – 2}{\sqrt{3x + 6}}, \quad a = 1 \]
77. \[ f(x) = \sqrt[3]{x – 1}, \quad a = -7 \]
78. \[ f(x) = \frac{(x – 1)^2}{(3x – 2)^2}, \quad a = \frac{1}{2} \]
79. \[ f(x) = \cot^2 x, \quad a = \frac{\pi}{4} \]
80. \[ f(x) = |1 – x^3|, \quad a = 2 \]
81. \[ f(x) = |\operatorname{sen} 5x|, \quad a = \frac{\pi}{3} \]

82. Hallar las rectas tangentes al gráfico de \( f(x) = (x – 1)(x – 2)(x – 3) \) en los puntos donde el gráfico corta al eje X.
83. Hallar los puntos en la gráfica de \( g(x) = x^2(x – 4)^2 \) en los cuales la recta tangente es paralela al eje X.
84. Hallar las rectas tangentes al gráfico de \( f(x) = \frac{x – 2}{x + 2} \) en los puntos donde este gráfico corta a los ejes. ¿Qué particularidad tienen estas rectas?
85. Hallar las rectas tangentes al gráfico de \( g(x) = \frac{x + 2}{x – 2} \) que pasan por el origen.
86. Hallar las rectas tangentes al gráfico de \( f(x) = 3x^2 – \ln x \) en el punto \( (1, 3) \).
87. Hallar las rectas tangentes al gráfico de \( y = \ln(1 + e^x) \) en el punto \( (0, \ln 2) \).
88. Sean \( f \) y \( g \) dos funciones diferenciables tales que \( f'(x) = \frac{1}{x} \) y \( f(g(x)) = x \). Probar que \( g'(x) = g(x) \).