f 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

f(x) = xg(x²)

f '(x) = g(x²) + x(2x)g '(x²) = g(x²) + 2x²g '(x²) Using product and chain rules

f "(x) = (2x)g '(x²) + 4xg '(x²) + 2x²(2x)g "(x²)

= 2xg '(x²) + 4xg '(x²) + 4x³g"(x²)

= 6g '(x²) + 4x³g "(x²)