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Let's go through the solution step by step.

### Given:
\[
y = (9 + 8x)^{5/x}
\]

#### a. Find \(\frac{dy}{dx}\) using logarithmic differentiation.
To use logarithmic differentiation, we start by taking the natural logarithm of both sides:
\[
\ln y = \ln\left((9 + 8x)^{5/x}\right)
\]
Using the power rule for logarithms, we can bring the exponent \( \frac{5}{x} \) down:
\[
\ln y = \frac{5}{x} \ln(9 + 8x)
\]

Now, differentiate both sides with respect to \(x\):
1. **Differentiate the left side** using implicit differentiation:
   \[
   \frac{1}{y} \frac{dy}{dx}
   \]
2. **Differentiate the right side** using the product rule:
   \[
   \frac{d}{dx}\left(\frac{5}{x} \ln(9 + 8x)\right) = \frac{5}{x} \cdot \frac{1}{9 + 8x} \cdot 8 - \frac{5 \ln(9 + 8

All parts of this problem refer to the function below:

y = (9 + 8x)^(5/x)

a. Use logarithmic differentiation to find dy/dx.
b. Find the slope of the tangent line at x = 1.

Learn how to use logarithmic differentiation to find dy/dx for the function y = (9 + 8x)^(5/x) and determine the slope of the tangent line at x = 1. This problem covers key calculus concepts, including implicit differentiation, the product rule, and logarithmic properties. Suitable for college-level calculus students.

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