Differentiate arcsin(2x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\sin^{-1}\left(2x\right)\right]$
2. 2. Step by step
   1. $$\frac{d}{dx}\left[\sin^{-1}\left(2x\right)\right] = \frac{2}{\sqrt{1-4{x}^{2}}}$$
      Chain rule — d/dx[arcsin u] = u′/√(1 − u²)
   2. $$\frac{d}{dx}\left[2x\right] = 2$$
      Constant multiple rule
   3. $$f'(x) = \frac{2}{\sqrt{1-4{x}^{2}}}$$
      Combine and simplify
3. 3. The answer
   $$f'(x) = \frac{2}{\sqrt{1-4{x}^{2}}}$$

   f'(x) = 2 / sqrt(1 - 4x²)

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

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