Find derivative of cos^-1 xFind derivative of cos^-1 x

The derivative of \( \cos^{-1}(x) \) (also written as \( \arccos(x) \)) with respect to \( x \) is:

\[
\frac{d}{dx} \left(\cos^{-1}(x)\right) = -\frac{1}{\sqrt{1 - x^2}}
\]

### Explanation:

The derivative is derived using the chain rule and the identity that relates the derivative of the inverse trigonometric functions. Specifically:

1. Start with \( y = \cos^{-1}(x) \), which implies \( x = \cos(y) \).
2. Differentiate both sides with respect to \( x \):
   \[
   \frac{d}{dx}(x) = \frac{d}{dx}(\cos(y))
   \]
   \[
   1 = -\sin(y) \cdot \frac{dy}{dx}
   \]
3. Solve for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = -\frac{1}{\sin(y)}
   \]
4. Since \( \sin(y) = \sqrt{1 - \cos^2(y)} = \sqrt{1 - x^2} \), we have:
   \[
   \frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}}
   \]

Thus, the derivative of \( \cos^{-1}(x) \) is \( -\frac{1}{\sqrt{1 - x^2}} \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the derivative of \( \sin^{-1}(x) \)?
2. How do you find the derivative of \( \tan^{-1}(x) \)?
3. What is the second derivative of \( \cos^{-1}(x) \)?
4. How is the chain rule applied in finding derivatives of inverse trigonometric functions?
5. How does the domain of \( \cos^{-1}(x) \) affect its derivative?

**Tip:** Remember that the domain of \( \cos^{-1}(x) \) is \( -1 \leq x \leq 1 \), which means the derivative is only defined within this interval.

Learn how to find the derivative of cos^-1 x (arccos(x)) with a step-by-step explanation. This problem involves calculus and inverse trigonometric functions, applying the chain rule for differentiation. Suitable for college-level mathematics.

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