Derivative of (sqrt(x^2+1))+(sqrt(3x-2)): animated lesson

1. 1. The problem
   Finding a derivative

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

   f'(x) = x / sqrt(x² + 1) + 3/2 / 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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