https://file.aiquickdraw.com/mathbot/file/1729618963498a1pooy5x.jpg

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.

Differentiate the following function with respect to x: y = e^(5x^2 + 3x - 7)

Learn how to differentiate the function e^(5x^2 + 3x - 7) using the chain rule in calculus. This step-by-step solution explains the application of the chain rule and provides insight into exponential functions and their derivatives.

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