Final answer:
To find f'(0) for the function f(x) = e⁵¹g(x), we apply the product rule and plug in the given values of g(0) and g'(0). The final result is f'(0) = 28.
Step-by-step explanation:
The function given is f(x) = e⁵¹g(x), and we are given g(0) = 6 and g'(0) = -2. To find f'(0), we need to apply the product rule for differentiation, which states that (uv)' = u'v + uv'. Let's set u = e⁵¹ and v = g(x); thus, u' = 5e⁵¹ and v' = g'(x).
Applying the product rule:
f'(x) = u'v + uv' = (5e⁵¹)g(x) + (e⁵¹)g'(x)
To find f'(0), plug in x = 0:
f'(0) = (5e⁵¹0)(g(0)) + (e⁵¹0)(g'(0))
= (5·1)(6) + (1)(-2)
= 30 - 2
= 28
Therefore, f'(0) = 28.