Найти производную функции

Найти производную функции

1. \( f'(x) = \frac{d}{dx}(4x^3 - 2x + 5) = 12x^2 - 2 \)
2. \( f'(x) = \frac{d}{dx}(x^2 \cdot \sin x) = 2x \sin x + x^2 \cos x \)
3. \( f'(x) = \frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2} \)
4. \( f'(x) = \frac{d}{dx}(e^x \cdot \cos x) = e^x \cos x - e^x \sin x \)
5. \( f'(x) = \frac{d}{dx}(\frac{x-3}{x+2}) = \frac{1(x+2) - 1(x-3)}{(x+2)^2} = \frac{x+2-x+3}{(x+2)^2} = \frac{5}{(x+2)^2} \)
6. \( f'(x) = \frac{d}{dx}(\sqrt{x^2}) = \frac{d}{dx}(|x|) \). Для \( x > 0 \), \( f'(x) = 1 \). Для \( x < 0 \), \( f'(x) = -1 \). В \( x=0 \) производная не существует.
7. \( f'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} \)
8. \( f'(x) = \frac{d}{dx}( g x) = \sec^2 x \)
9. \( f'(x) = \frac{d}{dx}((3x^2+1)^4) = 4(3x^2+1)^3 \cdot 6x = 24x(3x^2+1)^3 \)
10. \( f'(x) = \frac{d}{dx}(\sin(2x) \cdot e^x) = 2\cos(2x) \cdot e^x + \sin(2x) \cdot e^x = e^x(2\cos(2x) + \sin(2x)) \)