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Find \( f'(x) \) and \( f''(x) \) for \( f(x) = x^7 e^x \).

A) \( f'(x) = 7x^6 e^x + x^7 e^x \), \( f''(x) = 42x^5 e^x + 14x^6 e^x + x^7 e^x \)

B) \( f'(x) = 7x^6 e^x \), \( f''(x) = 42x^5 e^x + 14x^6 e^x \)

C) \( f'(x) = 7x^7 e^x + x^6 e^x \), \( f''(x) = 42x^5 e^x + 14x^6 e^x + x^7 e^x \)

D) \( f'(x) = 7x^7 e^x \), \( f''(x) = 42x^5 e^x + 14x^6 e^x + x^7 e^x \)

Answer :

Final answer:

To find f'(x) and f''(x) for the function f(x) = x⁷eˣ, we can use the product rule and chain rule of differentiation. f'(x) = 7x⁶eˣ + x⁷eˣ and f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ. The correct option is D) f'(x) = 7x⁷eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ

Explanation:

To find f'(x) and f''(x) for the function f(x) = x⁷eˣ, we can use the product rule and chain rule of differentiation.

First, let's find f'(x) (the first derivative):

  • Using the product rule, we have: f'(x) = (7x⁶)(eˣ) + (x⁷)(eˣ)
  • Simplifying, we get: f'(x) = 7x⁶eˣ + x⁷eˣ

Next, let's find f''(x) (the second derivative):

  • Using the product rule again, we have: f''(x) = (42x⁵)(eˣ) + (14x⁶)(eˣ) + (x⁷)(eˣ)
  • Simplifying, we get: f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ

The correct option is D) f'(x) = 7x⁷eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ

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