Final answer:
To find f'(4), we apply the product rule for differentiation to the function g(x)f(x). However, we do not have an explicit value for g'(4), and thus we cannot calculate an exact value for f'(4) without more information.
Step-by-step explanation:
The question asks us to find f'(4) given that g(x) · f(x) = 4 at x=4, g(4) = 2, and we have g'(4). To find f'(4), we apply the product rule for differentiation, which states that (uv)' = u'v + uv', where u and v are functions of x. In this case, u=g(x) and v=f(x).
Applying the product rule, we have:
(g(x)f(x))' = g'(x)f(x) + g(x)f'(x)
Since g(x)f(x) is a constant (4), its derivative with respect to x is 0. This gives us:
0 = g'(4)f(4) + g(4)f'(4)
Substituting the known values:
0 = g'(4)f(4) + 2f'(4)
We know that g(4) = 2, and g(x)f(x) = 4 at x=4, so we can solve for f(4):
f(4) = 4/g(4)
f(4) = 4/2
f(4) = 2
Now, we substitute f(4) back into the equation:
0 = g'(4)·(2) + 2f'(4)
However, we do not have a concrete value for g'(4), so we cannot calculate an exact value for f'(4). More information is needed to determine f'(4).