1. **State the problem:** Find the derivative of the function $$ f(x) = e^{\frac{3x - x^5}{6x^2}} $$ 2. **Rewrite the function:** Let $$ g(x) = \frac{3x - x^5}{6x^2} $$ so that $$ f(x) = e^{g(x)} = \exp(g(x)) $$ 3. **Apply chain rule:** The derivative of $f(x)$ is $$ f'(x) = e^{g(x)} \cdot g'(x) = \exp(g(x)) \cdot g'(x) $$ 4. **Find $g'(x)$:** Rewrite $g(x)$ as $$ g(x) = \frac{3x - x^5}{6x^2} = \frac{3x}{6x^2} - \frac{x^5}{6x^2} = \frac{3}{6x} - \frac{x^3}{6} = \frac{1}{2x} - \frac{x^3}{6} $$ 5. **Differentiate $g(x)$ term by term:** - Derivative of $\frac{1}{2x} = \frac{1}{2}x^{-1}$ is $$ \frac{d}{dx} \left( \frac{1}{2} x^{-1} \right) = \frac{1}{2} \cdot (-1)x^{-2} = -\frac{1}{2x^2} $$ - Derivative of $-\frac{x^3}{6}$ is $$ -\frac{1}{6} \cdot 3 x^2 = -\frac{1}{2} x^2 $$ 6. **Combine the derivatives:** $$ g'(x) = -\frac{1}{2x^2} - \frac{1}{2} x^2 = -\frac{1}{2x^2} - \frac{x^2}{2} $$ 7. **Write the final derivative:** $$ f'(x) = \exp\left( \frac{3x - x^5}{6x^2} \right) \cdot \left(-\frac{1}{2x^2} - \frac{x^2}{2} \right) $$ Or equivalently, $$ f'(x) = \exp\left( \frac{3x - x^5}{6x^2} \right) \cdot \left(-\frac{1}{2x^2} - \frac{x^2}{2}\right) $$ 8. **Express answer using the required format:** $$ f'(x) = \exp\left( \frac{3x - x^5}{6x^2} \right) \cdot \left(-\frac{1}{2x^2} - \frac{x^2}{2} \right) $$ or $$ f'(x) = \exp\left( \frac{3x - x^5}{6x^2} \right) \left(-\frac{1}{2*x^2} - \frac{x^2}{2}\right) $$ which in symbols is `exp((3*x - x^5)/(6*x^2))*(-1/(2*x^2) - x^2/2)`