**Calculus: Finding Derivatives****Problem Statement:**Find \( f'(x) \) and \( f''(x) \).Given:\[ f(x) = x^7 e^x \]**Solution:**1. **First Derivative \( f'(x) \):**\[ f'(x) = e^x x^6 (x + 7) \]The result has been boxed to indicate completion and correctness and is also marked with a green check mark.2. **Second Derivative \( f''(x) \):**A blank box is provided to input the second derivative.**Explanation:**- The function \( f(x) \) given is a product of \( x^7 \) and \( e^x \). To find the first derivative, we apply the product rule, which states: \[ (uv)' = u'v + uv' \]- Setting \( u = x^7 \) and \( v = e^x \), we find: \[ u' = 7x^6 \] \[ v' = e^x \]- Applying the product rule: \[ f'(x) = (x^7)'e^x + x^7(e^x)' \] \[ = 7x^6e^x + x^7e^x \] \[ = e^x (7x^6 + x^7) \] \[ = e^x x^6 (x + 7) \]- For the second derivative \( f''(x) \), you would need to apply the product and chain rules again.This transcription aims to provide clear and detailed steps for educational purposes, guiding users through the process of finding derivatives of a given function. Since the second derivative calculation is not completed in the image, practitioners are expected to apply their understanding of calculus to find \( f''(x) \).

Name: **Calculus: Finding Derivatives****Problem Statement:**Find \( f'(x)
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Calculus: Finding Derivatives****Problem Statement:**Find \( f'(x) \) and \( f''(x) \).Given:\[ f(x) = x^7 e^x \]**Solution:**1. **First Derivative \( f'(x) \):**\[ f'(x) = e^x x^6 (x + 7) \]The result has been boxed to indicate completion and cor