Derivative of (sqrt(3x-2))(cos(3x-2)): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\sqrt{3x-2} \cos\left(3x-2\right)\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[\sqrt{3x-2} \cos\left(3x-2\right)\right] = \frac{\frac{\cos\left(3x-2\right)\cdot3}{2}-3 {\sqrt{3x-2}}^{2} \sin\left(3x-2\right)}{\sqrt{3x-2}}$$
      Product rule: (uv)′ = u′v + uv′
   2. $$\frac{d}{dx}\left[\cos\left(3x-2\right)\right] = -\left(3\sin\left(3x-2\right)\right)$$
      Chain rule — d/dx[cos u] = −sin u · u′
   3. $$\frac{d}{dx}\left[\sqrt{3x-2}\right] = \frac{\frac{3}{2}}{\sqrt{3x-2}}$$
      Chain rule — d/dx[√u] = u′/(2√u)
   4. $$\frac{d}{dx}\left[3x-2\right] = 3$$
      Difference rule: differentiate each term separately
3. 3. Steps, part 2
   1. $$\frac{d}{dx}\left[3x\right] = 3$$
      Constant multiple rule
   2. $$f'(x) = \frac{\frac{\cos\left(3x-2\right)\cdot3}{2}-3 {\sqrt{3x-2}}^{2} \sin\left(3x-2\right)}{\sqrt{3x-2}}$$
      Combine and simplify
4. 4. The answer
   $$f'(x) = \frac{\frac{\cos\left(3x-2\right)\cdot3}{2}-3 {\sqrt{3x-2}}^{2} \sin\left(3x-2\right)}{\sqrt{3x-2}}$$

   f'(x) = (cos(3 x - 2)·3/2 - 3sqrt(3 x - 2)²·sin(3 x - 2)) / sqrt(3 x - 2)

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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