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The problem asks to estimate the slope \( m(6) \) of the function \( y = (2.3)^x \) at \( x = 6 \).

### To find the slope at a point:
The slope of the function at a specific point is found using the derivative of the function. The function is \( y = 2.3^x \). We'll first compute its derivative and then evaluate it at \( x = 6 \).

### Step 1: Differentiate \( y = 2.3^x \)

The derivative of an exponential function \( a^x \) with respect to \( x \) is given by:
\[
\frac{d}{dx} a^x = a^x \ln(a)
\]
For \( y = 2.3^x \), we have:
\[
\frac{dy}{dx} = 2.3^x \ln(2.3)
\]

### Step 2: Evaluate the derivative at \( x = 6 \)
We now substitute \( x = 6 \) into the derivative:
\[
m(6) = 2.3^6 \ln(2.3)
\]

Let's calculate this value to 2 decimal places.The slope of the function \( y = 2.3^x \) at \( x = 6 \), estimated to two decimal places, is approximately \( m(6) = 123.30 \).

Would you like more details or have any questions?

Here are five related questions to expand on this topic:
1. How do you find the derivative of other exponential functions like \( 3^x \)?
2. What is the significance of the natural logarithm in the derivative of an exponential function?
3. How would the slope change if the base of the exponential function were different, say \( 4^x \)?
4. How can the second derivative help in understanding the behavior of \( y = 2.3^x \)?
5. How would you calculate the slope for non-integer values of \( x \), like \( m(5.5) \)?

**Tip:** Exponential functions grow rapidly, and their derivatives reflect this by giving large slope values at higher \( x \)-values.

For any number r, let m(r) be the slope of the graph of the function y = (2.3)^x at the point x = r. Estimate m(6) to 2 decimal places.

Learn how to find the slope of the exponential function y = 2.3^x at x = 6 using derivatives. This involves calculating the derivative and evaluating it at the specific point. The solution is provided to 2 decimal places.

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