Derivative Calculator

Derivative of $\frac{1}{x}$

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

Step-by-step solution

Input function
$$\frac{1}{x}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\frac{1}{x}\right]}$$

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

Step 1 — Write the reciprocal as a negative power Rewriting
📖 Rule
$$\frac{c}{v}=c\cdot v^{-1},\quad \frac{c}{v^k}=c\cdot v^{-k}$$

Why: Write the denominator as a power with a negative exponent, then apply the factor and power rules. The original denominator must still be nonzero.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{x}\right]}\,}}}$$
✨ After: Write the reciprocal as a negative power
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{-1}\right]}\,}}}$$
Step 2 — Power Rule (special case) Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
With:
  • $u = x$
  • $c = -1$

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

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{-1}\right]}\,}}}$$
✨ After: Power Rule (special case)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1\,x^{-1 - 1}\,}}}$$
Step 3 — Evaluate numbers Simplification
📖 Rule

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

Changed part: $-1 - 1 \;\Longrightarrow\; -2$
🧮 Current expression
$$-1\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1 - 1\,}}}}$$
✨ After: Evaluate numbers
$$-1\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,}}}}$$
Step 4 — Write the factor −1 as a minus sign Simplification
📖 Rule
$$(-1)\cdot a=-a$$

Why: Multiplying by −1 gives the negative. Write just a minus sign instead.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1\,x^{-2}\,}}}$$
✨ After: Write the factor −1 as a minus sign
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-x^{-2}\,}}}$$
Step 5 — Write negative exponents as a fraction Simplification
📖 Rule
$$a^{-n}=\frac{1}{a^n}$$

Why: A power with a negative exponent is the reciprocal of the power with the positive exponent. The result then has no negative exponents.

Changed part: $x^{-2} \;\Longrightarrow\; \frac{1}{x^{2}}$
🧮 Current expression
$$-\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{-2}\,}}}$$
✨ After: Write negative exponents as a fraction
$$-\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2}}\,}}}$$
Final result
$$-\frac{1}{x^{2}}$$
Valid where:
  • $x\ne0$ – The original denominator must be nonzero.

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:

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