Differentiate x sin(x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[ x \sin\left( x\right)\right]$
2. 2. Step by step
   1. $$\frac{d}{dx}\left[ x \sin\left( x\right)\right] = \sin\left( x\right)+ x \cos\left( x\right)$$
      Product rule: (uv)′ = u′v + uv′
   2. $$\frac{d}{dx}\left[\sin\left( x\right)\right] = \cos\left( x\right)$$
      d/dx[sin x] = cos x
   3. $$f'(x) = \sin\left( x\right)+ x \cos\left( x\right)$$
      Combine and simplify
3. 3. The answer
   $$f'(x) = \sin\left( x\right)+ x \cos\left( x\right)$$

   f'(x) = sin(x) + x·cos(x)

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

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