Evaluate second derivative of ln(x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d^{2}}{dx^{2}}\left[\ln\left( x\right)\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[\ln\left( x\right)\right]$$
      Derivative number 1
   2. $$\frac{d}{dx}\left[\ln\left( x\right)\right] = \frac{1}{x}$$
      d/dx[ln x] = 1/x
   3. $$f'(x) = \frac{1}{x}$$
      Simplify the first derivative
   4. $$\frac{d}{dx}\left[\frac{1}{x}\right]$$
      Derivative number 2
3. 3. Steps, part 2
   1. $$\frac{d}{dx}\left[\frac{1}{x}\right] = -\frac{1}{{x}^{2}}$$
      Quotient rule: (u/v)′ = (u′v − uv′)/v²
   2. $$f''(x) = -\frac{1}{{x}^{2}}$$
      Simplify the second derivative
4. 4. The answer
   $$f''(x) = -\frac{1}{{x}^{2}}$$

   f''(x) = -(1 / x²)

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

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