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

Problem 2.314 · medium

1. \[ \frac{d}{d x} \frac{2 x}{2 x + 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = 2 \frac{d}{d x} \frac{x}{2 x + 1} \]
   constant-multipleFactor out the constant 2.✓ Proved
3. \[ = 2 x \frac{d}{d x} \frac{1}{2 x + 1} + \frac{2 \frac{d}{d x} x}{2 x + 1} \]
   productApply the product rule.✓ Proved
4. \[ = 2 x \frac{d}{d x} \frac{1}{2 x + 1} + \frac{2}{2 x + 1} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = - \frac{2 x \frac{d}{d x} \left(2 x + 1\right)}{\left(2 x + 1\right)^{2}} + \frac{2}{2 x + 1} \]
   chainApply the chain rule to the second term.✓ Proved
6. \[ = - \frac{4 x}{\left(2 x + 1\right)^{2}} + \frac{2}{2 x + 1} \]
   derivative algebra algebraDifferentiate the inner function 2*x + 1. Simplify the exponents and coefficients. Find a common denominator.✓ Proved
7. \[ = \frac{2}{\left(2 x + 1\right)^{2}} \]
   algebra simplify algebraCombine the fractions. Simplify the numerator. Final simplification.✓ Proved