differentiate ln(x^2+1) — step-by-step solution

Let's differentiate:

\frac{d}{d x} (\ln (x^{2} + 1))

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

x^{2} + 1

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (x^{2} + 1))

\frac{d}{d x} (x^{2} + 1) \cdot \frac{1}{x^{2} + 1}

\frac{d}{d x} (x^{2} + 1) \cdot \frac{1}{x^{2} + 1}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (x^{2} + 1) \cdot \frac{1}{x^{2} + 1}

\frac{\frac{d}{d x} (x^{2} + 1) \cdot 1}{x^{2} + 1}

\frac{\frac{d}{d x} (x^{2} + 1)}{x^{2} + 1}

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{2} + 1)

\frac{d}{d x} (x^{2}) + 0

\frac{\frac{d}{d x} (x^{2}) + 0}{x^{2} + 1}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (x^{2}) + 0

\frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

▶Derivative:

\frac{2 x}{x^{2} + 1}