Question

Let f(x) = eˣ g(x), where g(0) = 5 and g ′(0) = 2. find f ′(0).

Answer 1

To find f'(0) for function f(x) = e^x g(x), apply the product rule with values g(0) = 5 and g'(0) = 2, leading to the derivative f'(0) being 7.

Step-by-step explanation:

To find f'(0) for the function f(x) = exg(x), we can use the product rule for differentiation, which states that (u(x)v(x))' = u'(x)v(x) + u(x)v'(x). In this case, u(x) = ex and v(x) = g(x). The derivative of ex is ex itself, so u'(x) = ex, and v'(x) is simply g'(x).Using the given values of g(0) = 5 and g'(0) = 2, we can evaluate f'(x) at x = 0 as follows:

f'(0) = e0(g'(0)) + e0(g(0))

Since e0 is equal to 1, we have:

f'(0) = (1)(2) + (1)(5)

f'(0) = 2 + 5

f'(0) = 7.

Therefore, the derivative of f at x = 0, f'(0), is 7.