Derivative Calculator

Derivative of $\ln\left(x^{2} + 1\right)$

Differentiated with respect to x. Edit the function · Differentiation rules

Step-by-step solution

Input function
$$\ln\left(x^{2} + 1\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[\ln\left(x^{2} + 1\right)\right]}$$

Yellow: the expression being replaced. Green: its replacement.

Step 1 — Chain rule for the natural logarithm Differentiation rule
📖 Rule
$$\frac{d}{dx}\big(\ln({u})\big) = \frac{u'}{{u}}$$
With:
  • $u = x^{2} + 1$

Why: The outer derivative of the natural logarithm is the reciprocal of its argument. The chain rule adds the derivative of the argument in the numerator.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\ln\left(x^{2} + 1\right)\right]}\,}}}$$
✨ After: Chain rule for the natural logarithm
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[x^{2} + 1\right]}}{x^{2} + 1}\,}}}$$
Step 2 — Sum/Difference Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
With:
  • $u = x^{2}$
  • $v = 1$

Why: Differentiate every term separately and keep the plus and minus signs between them. Even a constant term first receives its own derivative symbol.

Changed part: $\frac{d}{dx}{\left[x^{2} + 1\right]} \;\Longrightarrow\; \frac{d}{dx}{\left[x^{2}\right]} + \frac{d}{dx}{\left[1\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2} + 1\right]}\,}}}}{x^{2} + 1}$$
✨ After: Sum/Difference Rule
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]} + \frac{d}{dx}{\left[1\right]}\,}}}}{x^{2} + 1}$$
Step 3 — Power Rule (special case) Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
With:
  • $u = x$
  • $c = 2$

Why: Bring the current exponent down as a multiplier and subtract one from the exponent. Evaluate that small calculation in a separate step.

Changed part: $\frac{d}{dx}{\left[x^{2}\right]} \;\Longrightarrow\; 2\,x^{2 - 1}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
✨ After: Power Rule (special case)
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
Step 4 — Evaluate numbers Simplification
📖 Rule

Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.

Changed part: $2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$\frac{2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
✨ After: Evaluate numbers
$$\frac{2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
Step 5 — Power with exponent one Simplification
📖 Rule
$$a^1=a$$

Why: An expression to the first power is the expression itself.

Changed part: $x^{1} \;\Longrightarrow\; x$
🧮 Current expression
$$\frac{2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
✨ After: Power with exponent one
$$\frac{2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}} + \frac{d}{dx}{\left[1\right]}}{x^{2} + 1}$$
Step 6 — Constant Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c)=0$$
With:
  • $c = 1$

Why: This term does not depend on x. Its value stays unchanged when x changes, so its derivative is zero. Other letters are treated as constant parameters.

Changed part: $\frac{d}{dx}{\left[1\right]} \;\Longrightarrow\; 0$
How does this rule work?
🧮 Current expression
$$\frac{2\,x + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1\right]}\,}}}}{x^{2} + 1}$$
✨ After: Constant Rule
$$\frac{2\,x + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}}{x^{2} + 1}$$
Step 7 — Remove a zero term Simplification
📖 Rule
$$a+0=a,\quad a-0=a$$

Why: Adding or subtracting zero leaves the value unchanged. Remove that zero term.

Changed part: $2\,x + 0 \;\Longrightarrow\; 2\,x$
🧮 Current expression
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x + 0\,}}}}{x^{2} + 1}$$
✨ After: Remove a zero term
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x\,}}}}{x^{2} + 1}$$
Final result
$$\frac{2\,x}{x^{2} + 1}$$
Valid where:
  • $x^{2} + 1>0$ – The logarithm argument must be positive.

The derivative is only valid where the original function is defined.

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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$\frac{2\,x}{x^{2} + 1}$$
Step-by-step result ↔ Maxima: identical ✓