Question

Let f(x) = ex g(x), where g(0) = 1 and g ′(0) = 2. Find f ′(0).

Answer 1

To find f'(0), we differentiated f(x) = e^x g(x) using the product rule and evaluated it at x = 0 with the given values of g(0) and g'(0), resulting in f'(0) = 3.

Step-by-step explanation:

To find f'(0), we need to differentiate f(x) = exg(x) and then evaluate it at x = 0.

Using the product rule for differentiation, we get:

f'(x) = exg'(x) + g(x)ex

Now, substituting the known values g(0) = 1 and g'(0) = 2:

f'(0) = e0(2) + (1)(e0)

Since e0 = 1, we simplify this to:

f'(0) = 2 + 1

Therefore, f'(0) = 3.