Differentiate y = ln(x^2 + 1).

Step 1 — Identify the outer and inner functions

The expression is a logarithm of a polynomial, so it is a composition:

• outer function: ln(u)
• inner function: u = x^2 + 1

Step 2 — Apply the chain rule

d/dx f(g(x)) = f'(g(x)) · g'(x)

The derivative of ln(u) with respect to u is 1/u, and du/dx = 2x.

Step 3 — Combine

dy/dx = (1 / (x^2 + 1)) · 2x = 2x / (x^2 + 1)

Check

The denominator x^2 + 1 is never zero, so the derivative is defined for every real x — a useful sanity check on logarithm problems, where a wrong inner function often produces a domain that does not match the original.