solve the slope of a tangent line at a given point            A, f(x)= $$ e^{\sqrt{x^2+1}} $$  , at  $$ x_o=\sqrt{3} $$

To solve for the slope of the tangent line to the function $ f(x) = e^{\sqrt{x^2 + 1}} $ at the point $ x_0 = \sqrt{3} $, we need to follow these steps:

### Step 1: Differentiate the function $ f(x) $
The slope of the tangent line is given by the derivative of the function, $ f'(x) $, evaluated at $ x_0 = \sqrt{3} $.

We begin by differentiating $ f(x) $:
\[
f(x) = e^{\sqrt{x^2 + 1}}
\]
We apply the chain rule:
1. The outer function is $ e^u $, where $ u = \sqrt{x^2 + 1} $. The derivative of $ e^u $ with respect to $ u $ is $ e^u $.
2. Now, differentiate $ u = \sqrt{x^2 + 1} $. Using the chain rule again:
   \[
   u = (x^2 + 1)^{1/2}
   \]
   The derivative of $ u $ with respect to $ x $ is:
   \[
   \frac{du}{dx} = \frac{1}{2}(x^2 + 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + 1}}
   \]

Thus, the derivative of $ f(x) $ is:
\[
f'(x) = e^{\sqrt{x^2 + 1}} \cdot \frac{x}{\sqrt{x^2 + 1}}
\]

### Step 2: Evaluate the derivative at $ x_0 = \sqrt{3} $
Now, we substitute $ x_0 = \sqrt{3} $ into $ f'(x) $:
\[
f'(\sqrt{3}) = e^{\sqrt{(\sqrt{3})^2 + 1}} \cdot \frac{\sqrt{3}}{\sqrt{(\sqrt{3})^2 + 1}}
\]
Simplify the terms:
\[
f'(\sqrt{3}) = e^{\sqrt{3^2 + 1}} \cdot \frac{\sqrt{3}}{\sqrt{3^2 + 1}} = e^{\sqrt{9 + 1}} \cdot \frac{\sqrt{3}}{\sqrt{9 + 1}} = e^{\sqrt{10}} \cdot \frac{\sqrt{3}}{\sqrt{10}}
\]

Thus, the slope of the tangent line at $ x_0 = \sqrt{3} $ is:
\[
f'(\sqrt{3}) = e^{\sqrt{10}} \cdot \frac{\sqrt{3}}{\sqrt{10}}
\]

### Final Answer:
The slope of the tangent line at $ x_0 = \sqrt{3} $ is:
\[
f'(\sqrt{3}) = e^{\sqrt{10}} \cdot \frac{\sqrt{3}}{\sqrt{10}}
\]

Would you like further details or clarification on any part of this solution?

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Here are 5 related questions:
1. How do you compute the derivative of composite functions using the chain rule?
2. What is the significance of the slope of a tangent line in calculus?
3. How do you differentiate functions involving square roots?
4. How do exponential functions behave when their exponents involve other functions?
5. What is the general formula for finding the derivative of a function at a given point?

**Tip:** Always simplify expressions involving square roots and exponents step by step to avoid mistakes.

Solve the slope of a tangent line at a given point A, f(x) = e^{sqrt(x^2+1)}, at x_0 = sqrt(3)

Discover how to find the slope of the tangent line for f(x) = e^{sqrt(x^2+1)} at the point x = sqrt(3) using calculus techniques, including differentiation and the chain rule. This problem is ideal for high school students in Grades 11-12.

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