Evaluate second derivative of x^4: animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d^{2}}{dx^{2}}\left[{x}^{4}\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[{x}^{4}\right]$$
      Derivative number 1
   2. $$\frac{d}{dx}\left[{x}^{4}\right] = 4{x}^{3}$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
   3. $$f'(x) = 4{x}^{3}$$
      Simplify the first derivative
   4. $$\frac{d}{dx}\left[4{x}^{3}\right]$$
      Derivative number 2
3. 3. Steps, part 2
   1. $$\frac{d}{dx}\left[4{x}^{3}\right] = 12{x}^{2}$$
      Constant multiple rule
   2. $$\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
   3. $$f''(x) = 12{x}^{2}$$
      Simplify the second derivative
4. 4. The answer
   $$f''(x) = 12{x}^{2}$$

   f''(x) = 12x²

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

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