g(x)>0 for all x and f(0)=1. if h(x)=f(x)g(x) and h'(x)=f(x)g'(x), then f(x)=?

Question

the answer is either:
a) f'(x)
b) g(x)
c) e^x
d) 0
e) 1

Answer 1

Remember the product rule of derivatives:
(uv)' = u'v + v'u

If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x).

Since the given h'x is missing the term f'(x)g(x), it means that f'(x)g(x) = 0. Since g(x) >0, then f'(x) must equal zero.

Thus, a-c can be ignored - since they don't provide an explanation for how f'(x) would equal 0.

f(x) can't equal 0, because f(0) = 1. That leaves answer e.

Answer 2

d) 0