The function you provided is:

$y = e^{5x^2 + 3x - 7}$

We are tasked with differentiating this function with respect to $x$.

Step-by-step solution:

1. Recognize the chain rule: Since the function is of the form $y = e^{u(x)}$, where $u(x) = 5x^2 + 3x - 7$, we apply the chain rule. The derivative of $e^{u(x)}$ is $e^{u(x)} \cdot \frac{du}{dx}$.

2. Differentiate $u(x) = 5x^2 + 3x - 7$: $\frac{du}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(3x) + \frac{d}{dx}(-7)$ $\frac{du}{dx} = 10x + 3$

3. Apply the chain rule: Now, the derivative of $y$ is: $\frac{dy}{dx} = e^{5x^2 + 3x - 7} \cdot (10x + 3)$

Thus, the derivative of the function is:

$\frac{dy}{dx} = e^{5x^2 + 3x - 7} \cdot (10x + 3)$

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

1. How do you apply the chain rule to more complex functions?
2. What is the derivative of $e^{x^3 + 2x^2}$?
3. How does the chain rule apply to trigonometric functions like $\sin(e^{2x})$?
4. How do you handle the product rule when combined with the chain rule?
5. What is the importance of exponential functions in calculus?

Tip: The chain rule is a powerful tool in differentiation. Practice applying it to various composite functions to build your intuition.