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 for clarity:** $$f(x) = e^{2x} \cdot x^{-1/2} \cdot (x^2+5)^{-1/4}$$ 3. **Use the product rule for derivatives:** If $$f(x) = u(x) \cdot v(x) \cdot w(x)$$, then $$f'(x) = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)$$. 4. **Identify each function:** - $$u(x) = e^{2x}$$ - $$v(x) = x^{-1/2}$$ - $$w(x) = (x^2+5)^{-1/4}$$ 5. **Find each derivative:** - $$u'(x) = 2e^{2x}$$ (chain rule) - $$v'(x) = -\frac{1}{2}x^{-3/2}$$ (power rule) - $$w'(x) = -\frac{1}{4}(x^2+5)^{-5/4} \cdot 2x = -\frac{1}{2}x(x^2+5)^{-5/4}$$ (chain rule) 6. **Substitute into the product rule formula:** $$f'(x) = 2e^{2x} \cdot x^{-1/2} \cdot (x^2+5)^{-1/4} + e^{2x} \cdot \left(-\frac{1}{2}x^{-3/2}\right) \cdot (x^2+5)^{-1/4} + e^{2x} \cdot x^{-1/2} \cdot \left(-\frac{1}{2}x(x^2+5)^{-5/4}\right)$$ 7. **Simplify each term:** - First term: $$2e^{2x} x^{-1/2} (x^2+5)^{-1/4}$$ - Second term: $$-\frac{1}{2} e^{2x} x^{-3/2} (x^2+5)^{-1/4}$$ - Third term: $$-\frac{1}{2} e^{2x} x^{1/2} (x^2+5)^{-5/4}$$ 8. **Factor out common terms:** $$f'(x) = e^{2x} x^{-3/2} (x^2+5)^{-5/4} \left[ 2x (x^2+5) + \left(-\frac{1}{2}\right)(x^2+5) - \frac{1}{2} x^2 \right]$$ 9. **Simplify inside the brackets:** $$2x(x^2+5) = 2x^3 + 10x$$ $$-\frac{1}{2}(x^2+5) = -\frac{1}{2}x^2 - \frac{5}{2}$$ $$-\frac{1}{2} x^2 = -\frac{1}{2} x^2$$ Sum: $$2x^3 + 10x - \frac{1}{2}x^2 - \frac{5}{2} - \frac{1}{2}x^2 = 2x^3 + 10x - x^2 - \frac{5}{2}$$ 10. **Final derivative:** $$f'(x) = e^{2x} x^{-3/2} (x^2+5)^{-5/4} \left( 2x^3 - x^2 + 10x - \frac{5}{2} \right)$$ This is the derivative of the given function.