Derivative Calculator

Derivative of $\arctan\left(x^{2}\right)$

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

Step-by-step solution

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

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

Step 1 — Derivative of Arctangent Differentiation rule
📖 Rule
$$\frac{d}{dx}\arctan\left(u\right)=\frac{1}{1 + u^{2}}\cdot u'$$
With:
  • $u = x^{2}$

Why: Differentiate the outer function while keeping its inner argument u. Multiply by u' and evaluate that inner derivative in its own step.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arctan\left(x^{2}\right)\right]}\,}}}$$
✨ After: Derivative of Arctangent
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{1 + \left(x^{2}\right)^{2}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
Step 2 — Power of a power Simplification
📖 Rule
$$\left(a^m\right)^n=a^{m\cdot n}$$

Why: Raising a power to another power multiplies the exponents.

Changed part: $\left(x^{2}\right)^{2} \;\Longrightarrow\; x^{2 \cdot 2}$
🧮 Current expression
$$\frac{1}{1 + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(x^{2}\right)^{2}\,}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
✨ After: Power of a power
$$\frac{1}{1 + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{2 \cdot 2}\,}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
Step 3 — Evaluate numbers Simplification
📖 Rule

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

Changed part: $2 \cdot 2 \;\Longrightarrow\; 4$
🧮 Current expression
$$\frac{1}{1 + x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 \cdot 2\,}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
✨ After: Evaluate numbers
$$\frac{1}{1 + x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,4\,}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
Step 4 — 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
$$\frac{1}{1 + x^{4}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
✨ After: Power Rule (special case)
$$\frac{1}{1 + x^{4}} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}$$
Step 5 — Evaluate numbers Simplification
📖 Rule

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

Changed part: $2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$\frac{1}{1 + x^{4}} \cdot 2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}$$
✨ After: Evaluate numbers
$$\frac{1}{1 + x^{4}} \cdot 2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}$$
Step 6 — 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
$$\frac{1}{1 + x^{4}} \cdot 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}$$
✨ After: Power with exponent one
$$\frac{1}{1 + x^{4}} \cdot 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}$$
Step 7 — Combine into one fraction Simplification
📖 Rule
$$a\cdot\frac{b}{c}=\frac{a\cdot b}{c}$$

Why: A factor in front of a fraction multiplies its numerator, giving a single fraction.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{1 + x^{4}} \cdot 2\,x\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{2\,x}{1 + x^{4}}\,}}}$$
Final result
$$\frac{2\,x}{1 + x^{4}}$$
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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$\frac{2\,x}{x^{4} + 1}$$
Step-by-step result ↔ Maxima: identical ✓