Derivative of $x^{2}$
Differentiated with respect to x. Edit the function · Differentiation rules
Step-by-step solution
Input function
$$x^{2}$$
Start of the differentiation
$$\frac{d}{dx}{\left[x^{2}\right]}$$
Yellow: the expression being replaced. Green: its replacement.
Step 1 — 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.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
✨ After: Power Rule (special case)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}$$
Step 2 — Evaluate numbers
Simplification
📖 Rule
Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.
Changed part:
$2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}$$
✨ After: Evaluate numbers
$$2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}$$
Step 3 — 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
$$2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}$$
✨ After: Power with exponent one
$$2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}$$
Final result
$$2\,x$$
Cross-check with a computer algebra system
The derivative computed directly by the computer algebra system Maxima:
$$2\,x$$
Step-by-step result ↔ Maxima: identical ✓