Find d/dx((3x+1)/(x-2)): animated lesson

1. 1. The problem
   Finding a derivative

   $\frac{d}{dx}\left[\frac{3x+1}{x-2}\right]$
2. 2. Steps, part 1
   1. $$\frac{d}{dx}\left[\frac{3x+1}{x-2}\right] = \frac{\frac{-\left(3x\right)-1}{x-2}+3}{x-2}$$
      Quotient rule: (u/v)′ = (u′v − uv′)/v²
   2. $$\frac{d}{dx}\left[ x-2\right] = 1$$
      Difference rule: differentiate each term separately
   3. $$\frac{d}{dx}\left[3x+1\right] = 3$$
      Sum rule: differentiate each term separately
   4. $$\frac{d}{dx}\left[3x\right] = 3$$
      Constant multiple rule
3. 3. Steps, part 2
   1. $$f'(x) = \frac{\frac{-\left(3x\right)-1}{x-2}+3}{x-2}$$
      Combine and simplify
4. 4. The answer
   $$f'(x) = \frac{\frac{-\left(3x\right)-1}{x-2}+3}{x-2}$$

   f'(x) = ((-3x - 1) / (x - 2) + 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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