Find the derivative of x/(x+1): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\frac{x}{x+1}\right]$
2. 2. Step by step
   1. $$\frac{d}{dx}\left[\frac{x}{x+1}\right] = \frac{1}{{\left( x+1\right)}^{2}}$$
      Quotient rule: (u/v)′ = (u′v − uv′)/v²
   2. $$\frac{d}{dx}\left[ x+1\right] = 1$$
      Sum rule: differentiate each term separately
   3. $$f'(x) = \frac{1}{{\left( x+1\right)}^{2}}$$
      Combine and simplify
3. 3. The answer
   $$f'(x) = \frac{1}{{\left( x+1\right)}^{2}}$$

   f'(x) = 1 / (x + 1)²

4. 4. Check
   • ✓Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

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