1. **State the problem:** Find the derivative of the function $$f(x) = \frac{e^{2x}}{x^{1/2}(x^2+5)^{1/4}}$$. 2. **Rewrite the function:** To differentiate, rewrite the function as $$f(x) = e^{2x} \cdot x^{-1/2} \cdot (x^2+5)^{-1/4}$$. 3. **Use the product rule:** Since $f(x)$ is a product of three functions, let $$u = e^{2x}, \quad v = x^{-1/2}, \quad w = (x^2+5)^{-1/4}$$. The derivative is $$f'(x) = u'vw + uv'w + uvw'$$. 4. **Find each derivative:** - $$u' = \frac{d}{dx} e^{2x} = 2e^{2x}$$ (chain rule) - $$v' = \frac{d}{dx} x^{-1/2} = -\frac{1}{2} x^{-3/2}$$ - $$w' = \frac{d}{dx} (x^2+5)^{-1/4} = -\frac{1}{4} (x^2+5)^{-5/4} \cdot 2x = -\frac{1}{2} x (x^2+5)^{-5/4}$$ (chain rule) 5. **Substitute back:** $$f'(x) = (2e^{2x}) \cdot x^{-1/2} \cdot (x^2+5)^{-1/4} + e^{2x} \cdot \left(-\frac{1}{2} x^{-3/2}\right) \cdot (x^2+5)^{-1/4} + e^{2x} \cdot x^{-1/2} \cdot \left(-\frac{1}{2} x (x^2+5)^{-5/4}\right)$$ 6. **Simplify terms:** $$f'(x) = 2 e^{2x} x^{-1/2} (x^2+5)^{-1/4} - \frac{1}{2} e^{2x} x^{-3/2} (x^2+5)^{-1/4} - \frac{1}{2} e^{2x} x^{1/2} (x^2+5)^{-5/4}$$ 7. **Final answer:** $$\boxed{f'(x) = e^{2x} \left[ 2 x^{-1/2} (x^2+5)^{-1/4} - \frac{1}{2} x^{-3/2} (x^2+5)^{-1/4} - \frac{1}{2} x^{1/2} (x^2+5)^{-5/4} \right]}$$ This derivative uses the product and chain rules carefully, and the function is expressed in a simplified form for clarity.