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To solve the problem, let's go through it step by step.

### 1. **Differentiate the function \( f(x) = x^7 \):**
   To find the derivative of \( f(x) = x^7 \), we use the power rule for differentiation. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is given by:

   \[
   f'(x) = nx^{n-1}
   \]

   Applying this to \( f(x) = x^7 \):

   \[
   f'(x) = 7x^{7-1} = 7x^6
   \]

### 2. **Calculate the slope of the graph at \( x = 2 \):**
   The slope of the graph of the function at a specific point is given by the value of the derivative at that point. Here, we need to find the slope at \( x = 2 \).

   Substitute \( x = 2 \) into the derivative \( f'(x) = 7x^6 \):

   \[
   f'(2) = 7(2)^6
   \]

   Now calculate \( 2^6 \):

   \[
   2^6 = 64
   \]

   So:

   \[
   f'(2) = 7 \times 64 = 448
   \]

### **Final Answer:**
The slope of the graph of \( y = x^7 \) at the point \( x = 2 \) is **448**.

Would you like further details or have any questions? Here are some related questions:

1. How does the power rule apply to functions with different powers?
2. What is the significance of finding the derivative of a function?
3. Can the same method be used to find the slope of the graph of any polynomial function?
4. How would the result change if the function were \( x^8 \) instead of \( x^7 \)?
5. What are the geometric implications of the slope of a function at a given point?

**Tip:** When dealing with higher powers, it's often useful to simplify calculations step by step to avoid errors, especially when substituting specific values.

Differentiate the function f(x) = x^7 and calculate the slope of the graph of y = x^7 at the point x = 2.

Learn how to differentiate the function f(x) = x^7 and calculate the slope of the graph at the point x = 2. This problem involves applying the power rule and evaluating the derivative at a specific point. Ideal for students in Grades 10-12.

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