Derivative Calculator

Derivative of $\sqrt{1 - x^{2}}$

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

Step-by-step solution

Input function
$$\sqrt{1 - x^{2}}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\sqrt{1 - x^{2}}\right]}$$

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

Step 1 — Derivative of the square root (chain rule) Differentiation rule
📖 Rule
$$\frac{d}{dx}\sqrt{u}=\frac{1}{2\,\sqrt{u}}\cdot u'$$
With:
  • $u = 1 - x^{2}$

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sqrt{1 - x^{2}}\right]}\,}}}$$
✨ After: Derivative of the square root (chain rule)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{2\,\sqrt{1 - x^{2}}}\,\frac{d}{dx}{\left[1 - x^{2}\right]}\,}}}$$
Step 2 — Sum/Difference Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
With:
  • $u = 1$
  • $v = x^{2}$

Why: Differentiate every term separately and keep the plus and minus signs between them. Even a constant term first receives its own derivative symbol.

Changed part: $\frac{d}{dx}{\left[1 - x^{2}\right]} \;\Longrightarrow\; \frac{d}{dx}{\left[1\right]} - \frac{d}{dx}{\left[x^{2}\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1 - x^{2}\right]}\,}}}$$
✨ After: Sum/Difference Rule
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(\frac{d}{dx}{\left[1\right]} - \frac{d}{dx}{\left[x^{2}\right]}\right)\,}}}$$
Step 3 — Constant Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(c)=0$$
With:
  • $c = 1$

Why: This term does not depend on x. Its value stays unchanged when x changes, so its derivative is zero. Other letters are treated as constant parameters.

Changed part: $\frac{d}{dx}{\left[1\right]} \;\Longrightarrow\; 0$
How does this rule work?
🧮 Current expression
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1\right]}\,}}} - \frac{d}{dx}{\left[x^{2}\right]}\right)$$
✨ After: Constant Rule
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}} - \frac{d}{dx}{\left[x^{2}\right]}\right)$$
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
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}\right)$$
✨ After: Power Rule (special case)
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}\right)$$
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
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - 2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\right)$$
✨ After: Evaluate numbers
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - 2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\right)$$
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
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}\right)$$
✨ After: Power with exponent one
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(0 - 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}\right)$$
Step 7 — Subtract from zero Simplification
📖 Rule
$$0-a=-a$$

Why: Zero minus an expression gives its negative.

Changed part: $0 - 2\,x \;\Longrightarrow\; -2\,x$
🧮 Current expression
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(0 - 2\,x\right)\,}}}$$
✨ After: Subtract from zero
$$\frac{1}{2\,\sqrt{1 - x^{2}}}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(-2\,x\right)\,}}}$$
Step 8 — Determine the sign of the product Simplification
📖 Rule
$$a\cdot(-b)=-(a\cdot b),\quad (-a)\cdot(-b)=a\cdot b$$

Why: Move the minus signs of the factors in front of the product. An even number of minus signs gives plus, an odd number gives minus.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{2\,\sqrt{1 - x^{2}}}\,\left(-2\,x\right)\,}}}$$
✨ After: Determine the sign of the product
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{2\,\sqrt{1 - x^{2}}} \cdot 2\,x\,}}}$$
Step 9 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{2\,\sqrt{1 - x^{2}}} \cdot 2\,x\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{2\,x}{2\,\sqrt{1 - x^{2}}}\,}}}$$
Step 10 — Cancel common numbers Simplification
📖 Rule
$$\frac{k\cdot a}{k\cdot b}=\frac{a}{b}$$

Why: Numerator and denominator share a numeric factor. Divide both by it.

Changed part: $\frac{2\,x}{2\,\sqrt{1 - x^{2}}} \;\Longrightarrow\; \frac{x}{\sqrt{1 - x^{2}}}$
🧮 Current expression
$$-\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{2\,x}{2\,\sqrt{1 - x^{2}}}\,}}}$$
✨ After: Cancel common numbers
$$-\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{x}{\sqrt{1 - x^{2}}}\,}}}$$
Final result
$$-\frac{x}{\sqrt{1 - x^{2}}}$$
Valid where:
  • $1 - x^{2}>0$ – The displayed root derivative applies on the positive branch; other real branches and endpoints require a separate check.

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:

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