Differentiate $\ln(x^2 + 1)$.

How we read your problem

Find the derivative of ln(x² + 1) with respect to x.

Step-by-step solution

1. 1

   Chain rule — d/dx[ln u] = u′/u

   $$\frac{d}{dx}\left[\ln\left({x}^{2}+1\right)\right] = \frac{2x}{{x}^{2}+1}$$
2. 2

   Sum rule: differentiate each term separately

   $$\frac{d}{dx}\left[{x}^{2}+1\right] = 2x$$
3. 3

   Power rule: d/dx[xⁿ] = n·xⁿ⁻¹

   $$\frac{d}{dx}\left[{x}^{2}\right] = 2x$$
4. 4

   Combine and simplify

   $$f'(x) = \frac{2x}{{x}^{2}+1}$$

Formulas used

• Power rule

  $$\frac{d}{dx}x^n = n x^{n-1}$$

• Chain rule

  $$\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$$

The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

Verification

• Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓