Derivative Calculator

Derivative of $\arcsin\left(\sqrt{x}\right)$

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

Step-by-step solution

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

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

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

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arcsin\left(\sqrt{x}\right)\right]}\,}}}$$
✨ After: Derivative of Arcsine
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - \left(\sqrt{x}\right)^{2}}}\,\frac{d}{dx}{\left[\sqrt{x}\right]}\,}}}$$
Step 2 — Square Root Rule (Simple) Differentiation rule
📖 Rule
$$\frac{d}{dx}\sqrt{x}=\frac{1}{2\,\sqrt{x}}$$

Why: The argument is exactly x. Apply the known elementary derivative.

Changed part: $\frac{d}{dx}{\left[\sqrt{x}\right]} \;\Longrightarrow\; \frac{1}{2\,\sqrt{x}}$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{1}{\sqrt{1 - \left(\sqrt{x}\right)^{2}}}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sqrt{x}\right]}\,}}}$$
✨ After: Square Root Rule (Simple)
$$\frac{1}{\sqrt{1 - \left(\sqrt{x}\right)^{2}}} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{2\,\sqrt{x}}\,}}}$$
Step 3 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - \left(\sqrt{x}\right)^{2}}} \cdot \frac{1}{2\,\sqrt{x}}\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - \left(\sqrt{x}\right)^{2}} \cdot 2\,\sqrt{x}}\,}}}$$
Step 4 — Order the factors Simplification
📖 Rule
$$u\cdot c=c\cdot u$$

Why: The order of factors may be changed (commutative law). Usual order: number, constants, powers of the variable, other factors.

Changed part: $\sqrt{1 - \left(\sqrt{x}\right)^{2}} \cdot 2\,\sqrt{x} \;\Longrightarrow\; 2\,\sqrt{x}\,\sqrt{1 - \left(\sqrt{x}\right)^{2}}$
🧮 Current expression
$$\frac{1}{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sqrt{1 - \left(\sqrt{x}\right)^{2}} \cdot 2\,\sqrt{x}\,}}}}$$
✨ After: Order the factors
$$\frac{1}{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sqrt{x}\,\sqrt{1 - \left(\sqrt{x}\right)^{2}}\,}}}}$$
Final result
$$\frac{1}{2\,\sqrt{x}\,\sqrt{1 - \left(\sqrt{x}\right)^{2}}}$$
Valid where:
  • $x>0$ – The displayed root derivative applies on the positive branch; other real branches and endpoints require a separate check.
  • $-1<\sqrt{x}<1$ – The derivative formula is applied in the interior of its real domain.

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

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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

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