Derivative Calculator

Derivative of $\arccos\left(\frac{1}{x}\right)$

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

Step-by-step solution

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

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

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

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arccos\left(\frac{1}{x}\right)\right]}\,}}}$$
✨ After: Derivative of Arccosine
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,\frac{d}{dx}{\left[\frac{1}{x}\right]}\,}}}$$
Step 2 — 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.

Changed part: $\frac{d}{dx}{\left[\frac{1}{x}\right]} \;\Longrightarrow\; \frac{d}{dx}{\left[x^{-1}\right]}$
🧮 Current expression
$$-\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{x}\right]}\,}}}$$
✨ After: Write the reciprocal as a negative power
$$-\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{-1}\right]}\,}}}$$
Step 3 — 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.

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

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

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

Why: Multiplication or division by one leaves the value unchanged.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,\left(-1\,x^{-2}\right)\,}}}$$
✨ After: Remove a factor of one
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,x^{-2}\,}}}$$
Step 6 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,x^{-2}\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{x^{-2}}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,}}}$$
Step 7 — 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
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{-2}\,}}}}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}$$
✨ After: Write negative exponents as a fraction
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2}}\,}}}}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}$$
Step 8 — Simplify a compound fraction Simplification
📖 Rule
$$\frac{a/b}{c}=\frac{a}{b\cdot c},\quad \frac{a}{b/c}=\frac{a\cdot c}{b}$$

Why: Dividing by a fraction means multiplying by its reciprocal, leaving a single fraction bar.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{1}{x^{2}}}{\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,}}}$$
✨ After: Simplify a compound fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2}\,\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}\,}}}$$
Final result
$$\frac{1}{x^{2}\,\sqrt{1 - \left(\frac{1}{x}\right)^{2}}}$$
Valid where:
  • $x\ne0$ – The original denominator must be nonzero.
  • $-1<\frac{1}{x}<1$ – The derivative formula is applied in the interior of its real domain.

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

Copy link to this solution
Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

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