Derivative of \( \displaystyle \sqrt{5} \sqrt{x} e^{5 x} \)

Problem 2.577 · hard

1. \[ \frac{d}{d x} \sqrt{5} \sqrt{x} e^{5 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \sqrt{5} \frac{d}{d x} \sqrt{x} e^{5 x} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \sqrt{5} \left(\sqrt{x} \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} \sqrt{x}\right) \]
   productApply the product rule.✓ Proved
4. \[ = \sqrt{5} \left(\sqrt{x} \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} x^{0.5}\right) \]
   rewriteRewrite sqrt(x) as x**0.5.✓ Proved
5. \[ = \sqrt{5} \left(\frac{0.5 e^{5 x}}{x^{0.5}} + \sqrt{x} \frac{d}{d x} e^{5 x}\right) \]
   derivativeDifferentiate x**0.5.✓ Proved
6. \[ = \sqrt{5} \left(\frac{0.5 e^{5 x}}{x^{0.5}} + 5 \sqrt{x} e^{5 x}\right) \]
   derivativeDifferentiate exp(5*x) using the chain rule.✓ Proved
7. \[ = \sqrt{5} \left(\frac{0.5}{x^{0.5}} + 5 \sqrt{x}\right) e^{5 x} \]
   algebraFactor out exp(5*x).✓ Proved
8. \[ = \sqrt{5} \left(5 \sqrt{x} + \frac{1}{2 \sqrt{x}}\right) e^{5 x} \]
   algebraRewrite 0.5*x**(-0.5) as 1/(2*sqrt(x)).✓ Proved
9. \[ = \frac{\sqrt{5} \left(10 x + 1\right) e^{5 x}}{2 \sqrt{x}} \]
   simplifyCombine the terms into a single fraction.✓ Proved