Derivative Calculator

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

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

Step-by-step solution

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

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:
  • $u = x^{2}$

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arccos\left(x^{2}\right)\right]}\,}}}$$
✨ After: Derivative of Arccosine
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - \left(x^{2}\right)^{2}}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
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
$$-\frac{1}{\sqrt{1 - \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(x^{2}\right)^{2}\,}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
✨ After: Power of a power
$$-\frac{1}{\sqrt{1 - \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{2 \cdot 2}\,}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
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
$$-\frac{1}{\sqrt{1 - x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 \cdot 2\,}}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
✨ After: Evaluate numbers
$$-\frac{1}{\sqrt{1 - x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,4\,}}}}}}\,\frac{d}{dx}{\left[x^{2}\right]}$$
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
$$-\frac{1}{\sqrt{1 - x^{4}}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
✨ After: Power Rule (special case)
$$-\frac{1}{\sqrt{1 - x^{4}}} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}$$
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
$$-\frac{1}{\sqrt{1 - x^{4}}} \cdot 2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}$$
✨ After: Evaluate numbers
$$-\frac{1}{\sqrt{1 - x^{4}}} \cdot 2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}$$
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
$$-\frac{1}{\sqrt{1 - x^{4}}} \cdot 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}$$
✨ After: Power with exponent one
$$-\frac{1}{\sqrt{1 - x^{4}}} \cdot 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}$$
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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - x^{4}}} \cdot 2\,x\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{2\,x}{\sqrt{1 - x^{4}}}\,}}}$$
Final result
$$-\frac{2\,x}{\sqrt{1 - x^{4}}}$$
Valid where:
  • $-1<x^{2}<1$ – The derivative formula is applied in the interior of its real domain.

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:

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