if f(x)=e^xg(x), where g(0)=1 and g’(0)=3, find f’(0)

Question

Finding answer for calculus homework

Answer 1

First, let's understand the question. It asks us to find f'(0), which means we must first take the derivative, and once we find its derivative, we plug in 0.

So, let's take the derivative. Notice that our function f is a product of two other functions involving x. Thus, to take the derivative, we need to use the product rule. Remember that (a(x) b(x))' = a(x)b'(x) + a'(x)b(x). The derivative of e^x is just e^x. The derivative of g(x) is g'(x). So,

f'(x) = e^x g'(x) + e^x g(x).

Now, we plug in. Remember that e⁰ = 1, and we are told g(0) = 1 and g'(0) = 3. So,

f'(0) = e⁰ * g'(0) + e⁰ + g(0) = 1 * 3 + 1 * 1 = 3 + 1 = 4.