1. Найдите производные функций f(x), если...

Решение:

1. \( \frac{d}{dx}(6x^{10} - 1) = 60x^9 \)
2. \( \frac{d}{dx}(12x^7 + 17x^3) = 84x^6 + 51x^2 \)
3. \( \frac{d}{dx}(11x^6 + 5x^{-24}) = 66x^5 - 120x^{-25} \)
4. \( \frac{d}{dx}((3x - 14)(3x^2 + 5)) = \frac{d}{dx}(9x^3 + 15x - 42x^2 - 70) = 27x^2 + 15 - 84x \)
5. \( \frac{d}{dx}(-3 \sin(5x - 6) + 12x^2) = -3 \cos(5x - 6) \cdot 5 + 24x = -15 \cos(5x - 6) + 24x \)
6. \( \frac{d}{dx}(\frac{4x - 11}{2 - 5x}) = \frac{4(2 - 5x) - (4x - 11)(-5)}{(2 - 5x)^2} = \frac{8 - 20x + 20x - 55}{(2 - 5x)^2} = \frac{-47}{(2 - 5x)^2} \)
7. \( \frac{d}{dx}(\frac{5x^2 - 2}{3x + 1}) = \frac{10x(3x + 1) - (5x^2 - 2)(3)}{(3x + 1)^2} = \frac{30x^2 + 10x - 15x^2 + 6}{(3x + 1)^2} = \frac{15x^2 + 10x + 6}{(3x + 1)^2} \)
8. \( \frac{d}{dx}(3 \sqrt{15 + 0.5x}) = \frac{d}{dx}(3(15 + 0.5x)^{1/2}) = 3 \cdot \frac{1}{2}(15 + 0.5x)^{-1/2} \cdot 0.5 = \frac{3}{4 \sqrt{15 + 0.5x}} \)