Find the derivative of 1/(x^2 + 1): animated lesson

1. 1. The problem
   Finding a derivative

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

   f'(x) = -(2x / (x² + 1)²)

4. 4. 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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