Derivative of \( \displaystyle \sqrt{2} \sqrt{x} e^{2 x} \)

Problem 2.591 · hard

1. \[ \frac{d}{d x} \sqrt{2} \sqrt{x} e^{2 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \sqrt{2} \frac{d}{d x} \sqrt{x} e^{2 x} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \sqrt{2} \left(\sqrt{x} \frac{d}{d x} e^{2 x} + e^{2 x} \frac{d}{d x} \sqrt{x}\right) \]
   productApply the product rule.✓ Proved
4. \[ = \sqrt{2} \left(\sqrt{x} \frac{d}{d x} e^{2 x} + e^{2 x} \frac{d}{d x} x^{0.5}\right) \]
   rewriteRewrite sqrt(x) as x**0.5.✓ Proved
5. \[ = \sqrt{2} \left(\frac{0.5 e^{2 x}}{x^{0.5}} + \sqrt{x} \frac{d}{d x} e^{2 x}\right) \]
   derivativeDifferentiate x**0.5.✓ Proved
6. \[ = \sqrt{2} \left(\frac{0.5 e^{2 x}}{x^{0.5}} + 2 \sqrt{x} e^{2 x}\right) \]
   derivative✓ Proved
7. \[ = \sqrt{2} \left(\frac{0.5}{x^{0.5}} + 2 \sqrt{x}\right) e^{2 x} \]
   algebraFactor out the common term exp(2*x).✓ Proved
8. \[ = \sqrt{2} \left(2 \sqrt{x} + \frac{1}{2 \sqrt{x}}\right) e^{2 x} \]
   algebraRewrite x**(-0.5) as 1/sqrt(x).✓ Proved
9. \[ = \frac{\sqrt{2} \left(4 x + 1\right) e^{2 x}}{2 \sqrt{x}} \]
   algebra simplifyCombine the terms inside the parentheses. Final simplified form.✓ Proved