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

Problem 2.12 · medium

1. \[ \frac{d}{d x} \frac{x}{x + 1} \]
   rewriteStart with the derivative of the function. Rewrite the fraction using a negative exponent.✓ Proved
2. \[ = x \frac{d}{d x} \frac{1}{x + 1} + \frac{\frac{d}{d x} x}{x + 1} \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} \frac{1}{x + 1} + \frac{1}{x + 1} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = - \frac{x \frac{d}{d x} \left(x + 1\right)}{\left(x + 1\right)^{2}} + \frac{1}{x + 1} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = - \frac{x}{\left(x + 1\right)^{2}} + \frac{1}{x + 1} \]
   derivative algebra rewrite algebraDifferentiate the inner function (x + 1). Simplify the expression. Rewrite the terms with positive exponents. Find a common denominator.✓ Proved
6. \[ = \frac{1}{\left(x + 1\right)^{2}} \]
   algebra simplifyCombine the numerators. Simplify the final expression.✓ Proved