Supply a valid technology formula for the given function.
(7 + x)^6x
x + 4

The function you provided looks like a composite expression, but it’s not clear if you're asking for simplification, differentiation, or some other operation involving technology formulas.

Given the expression \( (7 + x)^{6x} \) and the second term \( x + 4 \), I’ll provide some potential interpretations based on common requests:

### 1. Simplifying the expression:
There isn’t much that can be simplified directly between the terms \( (7 + x)^{6x} \) and \( x + 4 \). They are separate, and no simplifications can combine them.

### 2. Differentiating with respect to \( x \):
For differentiation, we can handle the two expressions separately:

- For the term \( (7 + x)^{6x} \), we use the **product rule** combined with the **chain rule** and **logarithmic differentiation**.
  
Let \( y = (7 + x)^{6x} \). Take the natural logarithm of both sides to apply logarithmic differentiation:

\[
\ln y = 6x \ln(7 + x)
\]

Differentiate both sides with respect to \( x \):

\[
\frac{dy}{dx} \cdot \frac{1}{y} = 6 \ln(7 + x) + 6x \cdot \frac{1}{7 + x}
\]

Multiply both sides by \( y = (7 + x)^{6x} \) to solve for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = (7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right)
\]

- For the second term \( x + 4 \), its derivative is straightforward:

\[
\frac{d}{dx} (x + 4) = 1
\]

So the derivative of the entire function is:

\[
\frac{d}{dx} \left[ (7 + x)^{6x} + (x + 4) \right] = (7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right) + 1
\]

This would be the formula for the derivative of the function with respect to \( x \).

### 3. Using in technology (computational tools):
If you need to input this into a calculator or software like WolframAlpha or any symbolic computing software (like Python’s SymPy or MATLAB), you would write:

```
f(x) = (7 + x)^(6x) + (x + 4)
diff(f(x), x)
```

This tells the system to differentiate the function \( f(x) \) with respect to \( x \).

### Summary:
The derivative of the function \( (7 + x)^{6x} + (x + 4) \) is:

\[
(7 + x)^{6x} \left( 6 \ln(7 + x) + \frac{6x}{7 + x} \right) + 1
\]

This is the most likely interpretation of your request.

Would you like more details or further clarifications?

### 5 Related Questions:
1. How do you apply the product rule to functions with exponents involving \( x \)?
2. What is logarithmic differentiation, and when is it useful?
3. Can you show how to solve the same function using a specific software like WolframAlpha or SymPy?
4. How does the chain rule apply in this problem?
5. What are the best strategies for simplifying complex exponential functions before differentiation?

### Tip:
When dealing with complex exponentials, applying logarithmic differentiation early on simplifies many steps, especially when both the base and exponent depend on \( x \).

Supply a valid technology formula for the given function: (7 + x)^6x and x + 4

This detailed solution covers the differentiation of the composite exponential function (7 + x)^(6x) + (x + 4) using the product rule, chain rule, and logarithmic differentiation. Suitable for advanced high school or early university students, the problem showcases key techniques in calculus.

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