The given function is:

$y = 5 + e^{4x + x^7}$

Let's find the derivative step by step.

Step 1: Differentiate constant term

The constant term "5" has a derivative of 0, so we can ignore it.

Step 2: Differentiate the exponential function

The function has the form $e^{u(x)}$, where $u(x) = 4x + x^7$. The derivative of $e^{u(x)}$ is:

$\frac{d}{dx} \left( e^{u(x)} \right) = e^{u(x)} \cdot \frac{d}{dx} (u(x))$

So we need to differentiate $u(x) = 4x + x^7$.

Step 3: Differentiate $u(x)$

The derivative of $u(x) = 4x + x^7$ is:

$\frac{d}{dx} (4x + x^7) = 4 + 7x^6$

Step 4: Apply the chain rule

Now, applying the chain rule, the derivative of the exponential term becomes:

$\frac{d}{dx} \left( e^{4x + x^7} \right) = e^{4x + x^7} \cdot (4 + 7x^6)$

Step 5: Final derivative

So, the derivative of the function is:

$y' = e^{4x + x^7} \cdot (4 + 7x^6)$

Let me know if you need further explanation or more steps detailed.

Here are some related questions:

1. How do you apply the chain rule in differentiation?
2. What are the properties of the exponential function in calculus?
3. How would you find the second derivative of this function?
4. What happens when the base of the exponential function is other than $e$?
5. Can you apply this method to logarithmic functions?

Tip: Always break down complex functions into smaller parts when differentiating, especially when you see nested functions.