differentiate\:\left(tan^{-1}\sqrt{x^2-1}+csc^{-1}x\right)

To differentiate the expression:

\[
f(x) = \tan^{-1}(\sqrt{x^2 - 1}) + \csc^{-1}(x)
\]

we need to differentiate each term individually.

### Step 1: Differentiate \( \tan^{-1}(\sqrt{x^2 - 1}) \)
Let \( u = \sqrt{x^2 - 1} \), so \( \tan^{-1}(u) \) is a composite function. Using the chain rule:
\[
\frac{d}{dx} \tan^{-1}(u) = \frac{1}{1+u^2} \cdot \frac{du}{dx}
\]
Now, differentiate \( u = \sqrt{x^2 - 1} \):
\[
\frac{du}{dx} = \frac{d}{dx} (x^2 - 1)^{1/2} = \frac{1}{2}(x^2 - 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 - 1}}
\]
Substituting \( u = \sqrt{x^2 - 1} \) and \( \frac{du}{dx} \) back:
\[
\frac{d}{dx} \left( \tan^{-1}(\sqrt{x^2 - 1}) \right) = \frac{1}{1 + (\sqrt{x^2 - 1})^2} \cdot \frac{x}{\sqrt{x^2 - 1}} = \frac{x}{(x^2)\sqrt{x^2 - 1}} = \frac{1}{x\sqrt{x^2 - 1}}
\]

### Step 2: Differentiate \( \csc^{-1}(x) \)
The derivative of \( \csc^{-1}(x) \) is given by:
\[
\frac{d}{dx} \csc^{-1}(x) = -\frac{1}{|x|\sqrt{x^2 - 1}}
\]

### Step 3: Combine the results
Now, we combine the derivatives of both terms:
\[
f'(x) = \frac{1}{x\sqrt{x^2 - 1}} - \frac{1}{|x|\sqrt{x^2 - 1}}
\]

For \( x > 1 \), where \( |x| = x \), the expression simplifies to:
\[
f'(x) = \frac{1}{x\sqrt{x^2 - 1}} - \frac{1}{x\sqrt{x^2 - 1}} = 0
\]

Thus, the derivative of the given expression is:
\[
f'(x) = 0 \text{ for } x > 1
\]

Would you like more details on any part of the calculation?

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Here are 5 questions that extend this discussion:
1. How do we handle derivatives of inverse trigonometric functions in general?
2. Can the derivative change if \( x \) is negative or less than 1?
3. How can we verify the behavior of \( f'(x) \) for different intervals of \( x \)?
4. Why does the expression simplify to zero in this particular case?
5. Can we apply similar techniques to differentiate products or quotients of trigonometric functions?

**Tip:** Always be cautious with the domain restrictions for inverse trigonometric functions to avoid errors in the final expression.

differentiate: tan^{-1}(√(x^2 - 1)) + csc^{-1}(x)

Explore how to differentiate tan^{-1}(√(x^2 - 1)) + csc^{-1}(x) using chain rule and inverse trigonometric function formulas. This detailed solution covers key calculus concepts, ideal for undergraduate students.

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