Find f'(x) and f"(x) where f(x) = x^4e^x

Question

asked Nov 9, 2024 37.3k views

1 Answer

Yavor Atov · Answer 1 · 2024-11-16T10:02:22+0000

Answer:

f'(x) =
${x^3}{e^x}(x + 4)$

f''(x) =
$2{x^2}{e^x}(2x^2 + 4x + 12)$

Explanation:

f(x) =
x^4 * e^x

Use the product rule:

(d)/(dx) (f(x) * g(x)) = f(x) * g'(x)+g(x) * f'(x))

x^4 * (d)/(dx) e^x+e^x * (d)/(dx) x^4

f'(x) =
x^4 * e^x + e^x * 4x^3

Use the sum rule:

(d)/(dx) (f(x) * g(x)) = f'(x) + g'(x)

(d)/(dx) (x^4 * e^x + e^x * 4x^3) = (d)/(dx) (x^4 * e^x) + (d)/(dx) (e^x * 4x^3)

Use the product rule for each part:

(d)/(dx) (f(x) * g(x)) = f(x) * g'(x)+g(x) * f'(x))

(d)/(dx) (x^4 * e^x) = x^4 * (d)/(dx) e^x + e^x * (d)/(dx) x^4

f''(x) (part one) =
x^4 * e^x + e^x * 4x^3

(d)/(dx) (x^4 * e^x) = x^4 * (d)/(dx) e^x + e^x * (d)/(dx) 4x^3

f''(x) (part two) =
x^4 * e^x + e^x * 12x^2

f''(x) =
x^4 * e^x + e^x * 4x^3 + x^4 * e^x + e^x * 12x^2

f''(x) =
${x^2}{e^x}(2x^2 + 4x + 12)$

f''(x) =
$2{x^2}{e^x}(x^2 + 2x + 6)$

So:

f'(x) =
x^4 * e^x + e^x * 4x^3

f''(x) = 2{x^2}{e^x}(x^2 + 2x + 6)