Find the function required to solve the equation

Question

What function would you take the derivative of to find the Maclaurin series of the function f(x)=4xe^{x^2}

Answer 1

4 x e^{x^2} = d/dx [ 2 e^{x^2} ] = 2 d/dx [ e^{x^2} ] = 2d/dx{ ∑_n=0 [ 1 + x² + x⁴ / 2! + x⁶ / 3 ! + x⁸ / 4 ! + x¹⁰ / 5 ! +....]}

= 2 ∑_n=0 [ 2x/1! + 4x³/2! + 6x⁵/ 3! + 8x⁷/4! + 10x⁹/5! +....] =

= 4 ∑_n=0 [ x/1! + 2x³/2! + 3x⁵/ 3! + 4x⁷/4! + 5x⁹/5! +....] =

= 4 ∑_n=0 [ x²ⁿ⁺¹/ n!]