1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{e^x}{x^3 + 2 \sin(x)}$$. 2. **Identify components:** This is a quotient of two functions: - Numerator: $u = e^x$ - Denominator: $v = x^3 + 2 \sin(x)$ 3. **Calculate derivatives:** - Derivative of the numerator: $$u' = \frac{d}{dx} e^x = e^x$$ - Derivative of the denominator: $$v' = \frac{d}{dx} (x^3 + 2 \sin(x)) = 3x^2 + 2 \cos(x)$$ 4. **Apply quotient rule:** The derivative of a quotient $\frac{u}{v}$ is given by $$f'(x) = \frac{u' * v - u * v'}{v^2}$$ 5. **Substitute values:** $$f'(x) = \frac{e^x * (x^3 + 2 \sin(x)) - e^x * (3x^2 + 2 \cos(x))}{(x^3 + 2 \sin(x))^2}$$ 6. **Factor out $e^x$ in the numerator:** $$f'(x) = \frac{e^x \left((x^3 + 2 \sin(x)) - (3x^2 + 2 \cos(x))\right)}{(x^3 + 2 \sin(x))^2}$$ 7. **Simplify the numerator:** $$f'(x) = \frac{e^x \left(x^3 + 2 \sin(x) - 3x^2 - 2 \cos(x)\right)}{(x^3 + 2 \sin(x))^2}$$ **Final answer:** $$f'(x) = \frac{e^x (x^3 + 2 \sin(x) - 3x^2 - 2 \cos(x))}{(x^3 + 2 \sin(x))^2}$$