Differentiate cos(x^2): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\cos\left({x}^{2}\right)\right]$
2. 2. Step by step
   1. $$\frac{d}{dx}\left[\cos\left({x}^{2}\right)\right] = -\left(2\sin\left({x}^{2}\right) x\right)$$
      Chain rule — d/dx[cos u] = −sin u · u′
   2. $$\frac{d}{dx}\left[{x}^{2}\right] = 2x$$
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
   3. $$f'(x) = -\left(2\sin\left({x}^{2}\right) x\right)$$
      Combine and simplify
3. 3. The answer
   $$f'(x) = -\left(2\sin\left({x}^{2}\right) x\right)$$

   f'(x) = -(2sin(x²)·x)

4. 4. 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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