Derivative of \( \displaystyle \sqrt{2} \sqrt{x} \ln{\left(2 x \right)} \)

Problem 2.485 · hard

1. \[ \frac{d}{d x} \sqrt{2} \sqrt{x} \ln{\left(2 x \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \sqrt{2} \frac{d}{d x} \sqrt{x} \ln{\left(2 x \right)} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \sqrt{2} \left(\sqrt{x} \frac{d}{d x} \ln{\left(2 x \right)} + \ln{\left(2 x \right)} \frac{d}{d x} \sqrt{x}\right) \]
   productApply the product rule.✓ Proved
4. \[ = \sqrt{2} \left(\sqrt{x} \frac{d}{d x} \ln{\left(2 x \right)} + \ln{\left(2 x \right)} \frac{d}{d x} x^{0.5}\right) \]
   rewriteRewrite sqrt(x) as a power.✓ Proved
5. \[ = \sqrt{2} \left(\frac{0.5 \ln{\left(2 x \right)}}{x^{0.5}} + \sqrt{x} \frac{d}{d x} \ln{\left(2 x \right)}\right) \]
   derivativeDifferentiate the power term.✓ Proved
6. \[ = \sqrt{2} \left(\frac{0.5 \ln{\left(2 x \right)}}{x^{0.5}} + \frac{\frac{d}{d x} 2 x}{2 \sqrt{x}}\right) \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = \sqrt{2} \left(\frac{0.5 \ln{\left(2 x \right)}}{x^{0.5}} + \frac{1}{\sqrt{x}}\right) \]
   derivative algebraDifferentiate the inner function 2x. Simplify the fraction.✓ Proved
8. \[ = \sqrt{2} \left(\frac{0.5 \ln{\left(2 x \right)}}{x^{0.5}} + x^{-0.5}\right) \]
   algebraSimplify sqrt(x)/x to x**-0.5.✓ Proved
9. \[ = \frac{\sqrt{2} \left(0.5 \ln{\left(2 x \right)} + 1\right)}{x^{0.5}} \]
   algebraFactor out x**-0.5.✓ Proved
10. \[ = \sqrt{2} \left(\frac{0.5 \ln{\left(2 x \right)}}{x^{0.5}} + x^{-0.5}\right) \]
    simplifyFinal simplified form.✓ Proved