Derivative of \( \displaystyle \frac{\sqrt{2} \sqrt{x}}{2 x + 1} \)

Problem 2.586 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{2} \sqrt{x}}{2 x + 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \sqrt{2} \frac{d}{d x} \frac{\sqrt{x}}{2 x + 1} \]
   constant-multiple rewritePull out the constant factor sqrt(2). Rewrite the denominator using a negative exponent.✓ Proved
3. \[ = \sqrt{2} \left(\sqrt{x} \frac{d}{d x} \frac{1}{2 x + 1} + \frac{\frac{d}{d x} \sqrt{x}}{2 x + 1}\right) \]
   productApply the product rule.✓ Proved
4. \[ = \sqrt{2} \left(- \frac{\sqrt{x} \frac{d}{d x} \left(2 x + 1\right)}{\left(2 x + 1\right)^{2}} + \frac{\frac{d}{d x} \sqrt{x}}{2 x + 1}\right) \]
   chainApply the chain rule to the second term.✓ Proved
5. \[ = \sqrt{2} \left(- \frac{2 \sqrt{x}}{\left(2 x + 1\right)^{2}} + \frac{\frac{d}{d x} \sqrt{x}}{2 x + 1}\right) \]
   derivative constant-multipleDifferentiate the inner function 2x + 1. Simplify the second term's constant.✓ Proved
6. \[ = \sqrt{2} \left(- \frac{2 \sqrt{x}}{\left(2 x + 1\right)^{2}} + \frac{1}{2 \sqrt{x} \left(2 x + 1\right)}\right) \]
   derivative algebraDifferentiate sqrt(x). Simplify the expression.✓ Proved
7. \[ = \frac{\sqrt{2} \left(1 - 2 x\right)}{2 \sqrt{x} \left(2 x + 1\right)^{2}} \]
   algebra algebra algebra simplifyFind a common denominator. Simplify the numerator. Combine like terms. Final simplified form.✓ Proved