Find d/dx((x^2 + 1)/(x - 1)): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\frac{{x}^{2}+1}{x-1}\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[\frac{{x}^{2}+1}{x-1}\right] = \frac{2x-\frac{{x}^{2}+1}{x-1}}{x-1}$$
      Quotient rule: (u/v)′ = (u′v − uv′)/v²
   2. $$\frac{d}{dx}\left[ x-1\right] = 1$$
      Difference rule: differentiate each term separately
   3. $$\frac{d}{dx}\left[{x}^{2}+1\right] = 2x$$
      Sum rule: differentiate each term separately
   4. $$\frac{d}{dx}\left[{x}^{2}\right] = 2x$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. 3. Steps, part 2
   1. $$f'(x) = \frac{2x-\frac{{x}^{2}+1}{x-1}}{x-1}$$
      Combine and simplify
4. 4. The answer
   $$f'(x) = \frac{2x-\frac{{x}^{2}+1}{x-1}}{x-1}$$

   f'(x) = (2x - (x² + 1) / (x - 1)) / (x - 1)

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

Differentiate with the power, product, quotient and chain rules, including higher derivatives.

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