We are given the function $f(x) = 1 + x \cdot e^{x^2 - 1}$, and we need to find the derivative of its inverse function $g(x)$ at $x = 1$, that is, $g'(1)$.

Step 1: Use the formula for the derivative of the inverse function

If $g$ is the inverse of $f$, then the derivative of the inverse function is given by the formula:

$g'(y) = \frac{1}{f'(g(y))}$

Thus, we need to find $g'(1)$, which can be written as:

$g'(1) = \frac{1}{f'(g(1))}$

Step 2: Find $g(1)$

To find $g(1)$, we need to solve for $x$ such that $f(x) = 1$. In other words, we need to solve the equation:

$1 + x \cdot e^{x^2 - 1} = 1$

Simplifying this:

$x \cdot e^{x^2 - 1} = 0$

This equation holds if $x = 0$, because for $x = 0$, $e^{x^2 - 1} = e^{-1}$, and the product $0 \cdot e^{-1} = 0$.

Thus, $g(1) = 0$.

Step 3: Find $f'(x)$

Next, we need to find the derivative of $f(x)$. Recall that $f(x) = 1 + x \cdot e^{x^2 - 1}$. To differentiate $f(x)$, we apply the product rule:

$f'(x) = \frac{d}{dx} \left( 1 + x \cdot e^{x^2 - 1} \right) = \frac{d}{dx} \left( x \cdot e^{x^2 - 1} \right)$

Using the product rule $\frac{d}{dx} (u \cdot v) = u' \cdot v + u \cdot v'$, where $u = x$ and $v = e^{x^2 - 1}$, we have:

$u' = 1 \quad \text{and} \quad v' = e^{x^2 - 1} \cdot 2x$

Thus,

$f'(x) = e^{x^2 - 1} + x \cdot e^{x^2 - 1} \cdot 2x = e^{x^2 - 1} \left( 1 + 2x^2 \right)$

Step 4: Evaluate $f'(g(1))$

Since $g(1) = 0$, we now evaluate $f'(0)$. From the expression for $f'(x)$, we have:

$f'(0) = e^{0^2 - 1} \left( 1 + 2 \cdot 0^2 \right) = e^{-1} \cdot 1 = \frac{1}{e}$

Step 5: Find $g'(1)$

Now we can use the formula for $g'(1)$:

$g'(1) = \frac{1}{f'(g(1))} = \frac{1}{f'(0)} = \frac{1}{\frac{1}{e}} = e$

Thus, $g'(1) = e$.

Final Answer:

$g'(1) = e$

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Would you like further details or clarification? Here are some related questions you might find interesting:

1. How would you find the derivative of $g(x)$ at other points?
2. What does the function $f(x) = 1 + x \cdot e^{x^2 - 1}$ represent geometrically?
3. Can we determine the behavior of $g(x)$ for large values of $x$?
4. What are the conditions under which the inverse function $g(x)$ exists?
5. How would you solve for $g(x)$ explicitly, if possible?

Tip: When dealing with inverse functions, always remember that the derivative of the inverse can be computed using the reciprocal of the derivative of the original function evaluated at the inverse point.