Evaluate second derivative of e^(2x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d^{2}}{dx^{2}}\left[{e}^{2x}\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[{e}^{2x}\right]$$
      Derivative number 1
   2. $$\frac{d}{dx}\left[{e}^{2x}\right] = 2{e}^{2x}$$
      Exponential rule: d/dx[eᵘ] = eᵘ·u′
   3. $$\frac{d}{dx}\left[2x\right] = 2$$
      Constant multiple rule
   4. $$f'(x) = 2{e}^{2x}$$
      Simplify the first derivative
3. 3. Steps, part 2
   1. $$\frac{d}{dx}\left[2{e}^{2x}\right]$$
      Derivative number 2
   2. $$\frac{d}{dx}\left[2{e}^{2x}\right] = 4{e}^{2x}$$
      Constant multiple rule
   3. $$\frac{d}{dx}\left[{e}^{2x}\right] = 2{e}^{2x}$$
      Exponential rule: d/dx[eᵘ] = eᵘ·u′
   4. $$\frac{d}{dx}\left[2x\right] = 2$$
      Constant multiple rule
4. 4. Steps, part 3
   1. $$f''(x) = 4{e}^{2x}$$
      Simplify the second derivative
5. 5. The answer
   $$f''(x) = 4{e}^{2x}$$

   f''(x) = 4e^(2x)

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

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