Derivative of ((2x)^3)+(ln(x^2+1)): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[{\left(2x\right)}^{3}+\ln\left({x}^{2}+1\right)\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[{\left(2x\right)}^{3}+\ln\left({x}^{2}+1\right)\right] = 24{x}^{2}+\frac{2x}{{x}^{2}+1}$$
      Sum rule: differentiate each term separately
   2. $$\frac{d}{dx}\left[\ln\left({x}^{2}+1\right)\right] = \frac{2x}{{x}^{2}+1}$$
      Chain rule — d/dx[ln u] = u′/u
   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. $$\frac{d}{dx}\left[{\left(2x\right)}^{3}\right] = 24{x}^{2}$$
      Power rule with the chain rule: d/dx[uⁿ] = n·uⁿ⁻¹·u′
   2. $$\frac{d}{dx}\left[2x\right] = 2$$
      Constant multiple rule
   3. $$f'(x) = 24{x}^{2}+\frac{2x}{{x}^{2}+1}$$
      Combine and simplify
4. 4. The answer
   $$f'(x) = 24{x}^{2}+\frac{2x}{{x}^{2}+1}$$

   f'(x) = 24x² + 2x / (x² + 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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