Derivative Calculator

Derivative of $\arccos\left(x\right) + \arcsin\left(x\right)$

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

Step-by-step solution

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

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

Step 1 — 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:
  • $u = \arccos\left(x\right)$
  • $v = \arcsin\left(x\right)$

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

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arccos\left(x\right) + \arcsin\left(x\right)\right]}\,}}}$$
✨ After: Sum/Difference Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arccos\left(x\right)\right]} + \frac{d}{dx}{\left[\arcsin\left(x\right)\right]}\,}}}$$
Step 2 — Derivative of Arccosine Differentiation rule
📖 Rule
$$\frac{d}{dx}\arccos\left(x\right)=-\frac{1}{\sqrt{1 - x^{2}}}$$

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

Changed part: $\frac{d}{dx}{\left[\arccos\left(x\right)\right]} \;\Longrightarrow\; -\frac{1}{\sqrt{1 - x^{2}}}$
How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arccos\left(x\right)\right]}\,}}} + \frac{d}{dx}{\left[\arcsin\left(x\right)\right]}$$
✨ After: Derivative of Arccosine
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - x^{2}}}\,}}} + \frac{d}{dx}{\left[\arcsin\left(x\right)\right]}$$
Step 3 — Derivative of Arcsine Differentiation rule
📖 Rule
$$\frac{d}{dx}\arcsin\left(x\right)=\frac{1}{\sqrt{1 - x^{2}}}$$

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

Changed part: $\frac{d}{dx}{\left[\arcsin\left(x\right)\right]} \;\Longrightarrow\; \frac{1}{\sqrt{1 - x^{2}}}$
How does this rule work? Wikipedia
🧮 Current expression
$$-\frac{1}{\sqrt{1 - x^{2}}} + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\arcsin\left(x\right)\right]}\,}}}$$
✨ After: Derivative of Arcsine
$$-\frac{1}{\sqrt{1 - x^{2}}} + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\sqrt{1 - x^{2}}}\,}}}$$
Step 4 — Opposite terms cancel Simplification
📖 Rule
$$u-u=0$$

Why: The same term appears once with plus and once with minus. Together they give zero, so both are removed.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{1}{\sqrt{1 - x^{2}}} + \frac{1}{\sqrt{1 - x^{2}}}\,}}}$$
✨ After: Opposite terms cancel
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}$$
Final result
$$0$$
Valid where:
  • $-1<x<1$ – The derivative formula is applied in the interior of its real domain.

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

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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$0$$
Step-by-step result ↔ Maxima: identical ✓