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We are given the function:

\[
f(x) = (4 + x)^6 + x
\]

We are asked to find the derivative \( f'(x) \).

### Step-by-step solution:

1. **Derivative of \( (4 + x)^6 \):**
   This is a composite function, so we will use the chain rule.

   The chain rule states that the derivative of \( (g(x))^n \) is:

   \[
   \frac{d}{dx}[(g(x))^n] = n \cdot (g(x))^{n-1} \cdot g'(x)
   \]

   For \( (4 + x)^6 \), we have:
   - \( g(x) = 4 + x \), so \( g'(x) = 1 \)
   - \( n = 6 \)

   Applying the chain rule:

   \[
   \frac{d}{dx}[(4 + x)^6] = 6 \cdot (4 + x)^5 \cdot 1 = 6(4 + x)^5
   \]

2. **Derivative of \( x \):**

   The derivative of \( x \) is simply 1.

3. **Combine the results:**

   Now, sum the derivatives:

   \[
   f'(x) = 6(4 + x)^5 + 1
   \]

### Final answer:

\[
f'(x) = 6(4 + x)^5 + 1
\]

---

Would you like more details on any of the steps? Here are some related questions for practice:

1. What is the derivative of \( (x^2 + 1)^7 \)?
2. How would you apply the chain rule to \( \sin(3x) \)?
3. What is the derivative of \( (2x + 5)^4 \)?
4. How does the chain rule apply to functions like \( \ln(4x + 1) \)?
5. Find the second derivative of \( f(x) = (4 + x)^6 + x \).

**Tip:** When using the chain rule, always differentiate the "inner" and "outer" functions separately and then multiply them together.

Find the derivative of the function f(x) = (4 + x)^6 + x, x > -4. Express f'(x) in terms of x.

Learn how to differentiate the function f(x) = (4 + x)^6 + x with a detailed step-by-step solution. This problem covers key calculus concepts like the chain rule and is suitable for students in Grades 10-12.

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