To find $g'(1)$, where $g(x)$ is the inverse function of $f(x) = \ln(2x - 3)$, we can use two methods. Let’s proceed step by step:

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1. Using the Relationship Between a Function and Its Inverse

The derivative of an inverse function satisfies: $g'(a) = \frac{1}{f'(g(a))},$ where $f(g(a)) = a$.

Step 1.1: Determine $g(1)$

Since $g(x)$ is the inverse of $f(x)$, we solve $f(x) = 1$ to find the value of $g(1)$: $f(x) = \ln(2x - 3) = 1.$ Exponentiating both sides: $2x - 3 = e^1 \quad \Rightarrow \quad 2x = e + 3 \quad \Rightarrow \quad x = \frac{e + 3}{2}.$ Thus, $g(1) = \frac{e + 3}{2}$.

Step 1.2: Compute $f'(x)$

The derivative of $f(x)$ is: $f'(x) = \frac{d}{dx} \ln(2x - 3) = \frac{1}{2x - 3} \cdot 2 = \frac{2}{2x - 3}.$

Step 1.3: Evaluate $f'(g(1))$

Substitute $g(1) = \frac{e + 3}{2}$ into $f'(x)$: $f'(g(1)) = \frac{2}{2\left(\frac{e + 3}{2}\right) - 3} = \frac{2}{e + 3 - 3} = \frac{2}{e}.$

Step 1.4: Compute $g'(1)$

Using $g'(1) = \frac{1}{f'(g(1))}$: $g'(1) = \frac{1}{\frac{2}{e}} = \frac{e}{2}.$

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2. Directly Differentiating the Inverse Relationship

The functions $f(x)$ and $g(x)$ satisfy: $f(g(x)) = x.$ Differentiating both sides with respect to $x$: $f'(g(x)) \cdot g'(x) = 1.$ Thus: $g'(x) = \frac{1}{f'(g(x))}.$

Step 2.1: Evaluate $g(1)$

As determined earlier, $g(1) = \frac{e + 3}{2}$.

Step 2.2: Compute $f'(g(1))$

From earlier: $f'(g(1)) = \frac{2}{e}.$

Step 2.3: Compute $g'(1)$

$g'(1) = \frac{1}{f'(g(1))} = \frac{1}{\frac{2}{e}} = \frac{e}{2}.$

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Final Answer:

$g'(1) = \frac{e}{2}.$

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Do you want me to expand further on any step or concept?
Here are 5 follow-up questions related to this problem:

1. What is the geometric interpretation of the derivative of an inverse function?
2. How does the chain rule apply in differentiating $f(g(x)) = x$?
3. What are the conditions for the existence of an inverse function for $f(x)$?
4. How do we find the domain and range of $f(x) = \ln(2x - 3)$?
5. Can you compute $g'(x)$ for a general $x$ instead of at a specific point?

Tip: Remember that the derivative of an inverse function at a point can be interpreted as the reciprocal of the slope of the original function at the corresponding point.