Question

Find f '(x) and f ''(x). f(x) = x⁷eˣ A) f'(x) = 7x⁶eˣ + x⁷eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ

B) f'(x) = 7x⁶eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ
C) f'(x) = 7x⁷eˣ + x⁶eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ
D) f'(x) = 7x⁷eˣ, f''(x) = 42x⁵eˣ + 14x⁶eˣ + x⁷eˣ

Answer 1

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

Step-by-step 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