1. Dada la función $$f(x) = 2x^6 + 3x^5 - 7x^4 + 9x^2 - 8x - 26$$, calculamos la cuarta derivada $$f^{(iv)}(x)$$: 1. Derivada 1: $$f'(x) = 12x^5 + 15x^4 - 28x^3 + 18x - 8$$ 2. Derivada 2: $$f''(x) = 60x^4 + 60x^3 - 84x^2 + 18$$ 3. Derivada 3: $$f'''(x) = 240x^3 + 180x^2 - 168x$$ 4. Derivada 4: $$f^{(iv)}(x) = 720x^2 + 360x - 168$$ 2. Para $$F(x) = 3x^{-4} - 2x^{-2} + 13x + 8x^3$$, calculamos la tercera derivada $$F'''(x)$$: 1. Derivada 1: $$F'(x) = -12x^{-5} + 4x^{-3} + 13 + 24x^2$$ 2. Derivada 2: $$F''(x) = 60x^{-6} - 12x^{-4} + 48x$$ 3. Derivada 3: $$F'''(x) = -360x^{-7} + 48x^{-5} + 48$$ 3. Para $$g(x) = \ln(6x^2) + 13ab^2$$, las constantes desaparecen al derivar respecto a $$x$$. Derivamos tres veces: 1. Derivada 1: $$g'(x) = \frac{12x}{6x^2} = \frac{2}{x}$$ 2. Derivada 2: $$g''(x) = -\frac{2}{x^2}$$ 3. Derivada 3: $$g'''(x) = \frac{4}{x^3}$$ 4. Para $$h(x) = e^{5x - 2}$$, calculamos la cuarta derivada $$h^{(iv)}(x)$$: Cada derivada agrega un factor 5 por la regla de cadena: 1. $$h'(x) = 5 e^{5x - 2}$$ 2. $$h''(x) = 25 e^{5x - 2}$$ 3. $$h'''(x) = 125 e^{5x - 2}$$ 4. $$h^{(iv)}(x) = 625 e^{5x - 2}$$ 5. Para $$j(x) = 2 \tan(3x + 1)$$, calculamos la segunda derivada $$j''(x)$$: 1. Derivada 1: $$j'(x) = 2 \cdot 3 \sec^{2}(3x + 1) = 6 \sec^{2}(3x + 1)$$ 2. Derivada 2 usando la regla de la cadena: $$j''(x) = 6 \cdot 2 \sec^{2}(3x + 1) \tan(3x + 1) \cdot 3 = 36 \sec^{2}(3x + 1) \tan(3x + 1)$$ 6. Para $$k(x) = x^{2} \cos x$$, calculamos la tercera derivada $$k'''(x)$$ usando la regla del producto repetidamente: 1. $$k'(x) = 2x \cos x - x^{2} \sin x$$ 2. $$k''(x) = 2 \cos x - 2x \sin x - 2x \sin x - x^{2} \cos x = 2 \cos x - 4x \sin x - x^{2} \cos x$$ 3. $$k'''(x) = -2 \sin x -4 \sin x - 4x \cos x - 2x \cos x + x^{2} \sin x = -6 \sin x - 6x \cos x + x^{2} \sin x$$ 7. Para $$m(x) = 3x \sin(4x)$$, calculamos la segunda derivada $$m''(x)$$: 1. $$m'(x) = 3 \sin(4x) + 3x \cdot 4 \cos(4x) = 3 \sin(4x) + 12x \cos(4x)$$ 2. $$m''(x) = 12 \cos(4x) + 12 \cos(4x) - 48x \sin(4x) = 24 \cos(4x) - 48x \sin(4x)$$ 8. Para $$n(x) = 5 \sec(2x)$$, calculamos la segunda derivada $$n''(x)$$: 1. $$n'(x) = 5 \cdot 2 \sec(2x) \tan(2x) = 10 \sec(2x) \tan(2x)$$ 2. $$n''(x) = 10 [2 \sec(2x) \tan^{2}(2x) + 2 \sec^{3}(2x)] = 20 \sec(2x) \tan^{2}(2x) + 20 \sec^{3}(2x)$$ 9. Para $$y = \sin^{4}(2x)$$, calculamos la tercera derivada $$y'''(x)$$: Sea $$u=\sin(2x)$$, entonces $$y = u^{4}$$. 1. $$y' = 4 u^{3} u' = 4 \sin^{3}(2x) \cdot 2 \cos(2x) = 8 \sin^{3}(2x) \cos(2x)$$ 2. Derivando $$y'$$ usando producto y cadena para obtener $$y''$$: $$y'' = 8 [3 \sin^{2}(2x) \cos(2x) \cdot 2 \cos(2x) + \sin^{3}(2x)(-2 \sin(2x))] = 48 \sin^{2}(2x) \cos^{2}(2x) - 16 \sin^{4}(2x)$$ 3. Derivando para $$y'''$$: $$y''' = 48 [2 \sin(2x) \cos^{3}(2x) \cdot 2 + \sin^{2}(2x) \cdot 3 \cos^{2}(2x)(-2 \sin(2x))] - 64 \sin^{3}(2x) \cos(2x)$$ Simplificando se obtiene : $$y''' = 192 \sin(2x) \cos^{3}(2x) - 288 \sin^{3}(2x) \cos(2x)$$ 10. Para $$p(x) = e^{3x^{2}}$$, calculamos la tercera derivada $$p'''(x)$$: 1. $$p'(x) = e^{3x^{2}} \cdot 6x = 6x e^{3x^{2}}$$ 2. $$p''(x) = 6 e^{3x^{2}} + 6x \cdot e^{3x^{2}} \cdot 6x = 6 e^{3x^{2}} + 36 x^{2} e^{3x^{2}} = (6 + 36 x^{2}) e^{3x^{2}}$$ 3. $$p'''(x) = (72 x) e^{3x^{2}} + (6 + 36 x^{2}) \cdot 6x e^{3x^{2}} = (72 x + 36 x + 216 x^{3}) e^{3x^{2}} = (108 x + 216 x^{3}) e^{3x^{2}}$$