Question

What is the first and second derivative of xe^(2x)?

Answer 1

`f(x)=xe^(2x)\\\\f'(x)=x'e^(2x)+x(e^(2x))'=e^(2x)+xe^(2x)\cdot2=2xe^(2x)+e^(2x)=e^(2x)(2x+1)\\\\f''(x)=(e^(2x))'(2x+1)+e^(2x)(2x+1)'=e^(2x)\cdot2(2x+1)+e^(2x)\cdot2\\\\=e^(2x)(4x+2)+2e^(2x)=(4x+2+2)e^(2x)=(4x+4)e^(2x)\\\\Answer:\\\boxed{f'(x)=(2x+1)e^(2x)\ and\ f''(x)=(4x+4)e^(2x)}`

Answer 2

The derivative of a function xe^(2x) can be determined by applying differential calculus techniques to the equation. In this case, the applicable law is the law of products

y = xe^(2x)
y' = 2x e^2x + e^2x
y'' = 4x e^2x + 2e^2x + 2 e^2x
Simplifying
y'' = (4x+4) e^2x