To differentiate the function:

$f(x) = e^{x^3 + 5x}$

We need to apply the chain rule, as we have an exponential function with a more complex exponent $x^3 + 5x$.

Steps:

1. The derivative of $e^u$ with respect to $u$ is $e^u$, where $u = x^3 + 5x$.
2. Now, differentiate the exponent $u = x^3 + 5x$ with respect to $x$.

$\frac{d}{dx}(x^3 + 5x) = 3x^2 + 5$

3. Apply the chain rule: multiply $e^{x^3 + 5x}$ by the derivative of $x^3 + 5x$:

$f'(x) = e^{x^3 + 5x} \cdot (3x^2 + 5)$

Final Answer:

$f'(x) = e^{x^3 + 5x} \cdot (3x^2 + 5)$

Would you like more details, or do you have any questions?

Here are 5 related questions you might find helpful:

1. How do you differentiate functions involving products or quotients?
2. What is the chain rule in differentiation, and how does it work?
3. How do you handle logarithmic functions when differentiating?
4. What are higher-order derivatives, and how do you calculate them?
5. How does implicit differentiation differ from explicit differentiation?

Tip: When dealing with exponential functions, always check whether the exponent is a function of $x$ and use the chain rule if necessary.