Differentiate x^3 cos(2x): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[{x}^{3} \cos\left(2x\right)\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[{x}^{3} \cos\left(2x\right)\right] = 3{x}^{2} \cos\left(2x\right)-2{x}^{3} \sin\left(2x\right)$$
      Product rule: (uv)′ = u′v + uv′
   2. $$\frac{d}{dx}\left[\cos\left(2x\right)\right] = -\left(2\sin\left(2x\right)\right)$$
      Chain rule — d/dx[cos u] = −sin u · u′
   3. $$\frac{d}{dx}\left[2x\right] = 2$$
      Constant multiple rule
   4. $$\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. 3. Steps, part 2
   1. $$f'(x) = 3{x}^{2} \cos\left(2x\right)-2{x}^{3} \sin\left(2x\right)$$
      Combine and simplify
4. 4. The answer
   $$f'(x) = 3{x}^{2} \cos\left(2x\right)-2{x}^{3} \sin\left(2x\right)$$

   f'(x) = 3x²·cos(2x) - 2x³·sin(2x)

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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