Question

Let f and g be two functions whose second derivatives are defined. Then (f · g) '' = f · g '' + f '' · g?

Answer 1

\[\large{(f \cdot g) '' \\
= {d\over dx}\left( (f \cdot g)' \right) \\
= {d\over dx}(f'g + fg') \\
= {d\over dx}(f'g) + {d\over dx}(fg') \\
= (f')'(g) +(f')(g)' \ \ \ \ \ + \ \ \ \ \ (f)'(g') + (f)(g')' \\
= f''g + 2f'g' + fg'' }\]

Answer 2

2f'g' is probably not 0, so the statement is false.