Natural Logarithm Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Find the derivative of

f(x) = \ln(x^2 + 1)

and determine where

f

is increasing.

1
By the chain rule: $f'(x) = \frac{1}{x^2+1} \cdot 2x = \frac{2x}{x^2+1}$ .
2
$f'(x) > 0$ when $2x > 0$ (since $x^2+1 > 0$ always), i.e., when $x > 0$ . So $f$ is increasing on $(0, \infty)$ .

Answer

f'(x) = \frac{2x}{x^2+1}, \quad f \text{ is increasing on } (0, \infty)

The derivative of

\ln(u)

\frac{u'}{u}

by the chain rule. Since

x^2+1

is always positive, the sign of

f'(x)

depends entirely on the numerator

2x

. This shows

\ln

composed with a function inherits increasing/decreasing behavior from the argument's derivative.

About Natural Logarithm

The logarithm with base $e \approx 2.71828$ : $\ln x = \log_e x$ . It is the inverse function of $e^x$ .

Evaluate [formula].

Simplify [formula].

Solve [formula] for [formula].