If g is twice differentiable function and ...

Question

If g is twice differentiable function and f(x) = xg(x2) find f′′(x) in terms of g,g′, ′′

and g . Simplify your answer as much as possible. (Hint: to say that g is twice differentiable means both g′ and g′′ exist.)

Answer 1

Let f(x) =x g(x²). So f'(x) = g(x²) + 2x²g'(x²). f"(x)= 2xg'(x²) +4x g'(x²) +4 x³ g"(x²). So f"(x)= 6 x g'(x²) + 4 x³ g"(x²).