Answer to Question #275760 in Calculus for JAY

Question

1. D(x²+2x-3)                  ans, 2x+2
2. D²(xe^3x-e^4x)                 ans, 9xe^3x+63^3x-16e^4x

Answer 1

$1. \ D(x^2+2x-3)=Dx^2+D(2x-3)=2x+2$

$2. \ D(xe^{3x}-e^{4x})=D(xe^{3x})-De^{4x}=Dx\cdot e^{3x}+x\cdot De^{3x}-De^{4x}= \\ =e^{3x}+x\cdot e^{3x}\cdot D(3x)-e^{4x}\cdot D(4x)=e^{3x}+3xe^{3x}-4e^{4x}$

$D^2(xe^{3x}-e^{4x})=D((1+3x)e^{3x}-4e^{4x})=D(1+3x)\cdot e^{3x}+(1+3x)\cdot De^{3x}-4De^{4x}=3e^{3x}+(1+3x)\cdot 3e^{3x}-16e^{4x}=9xe^{3x}+6e^{3x}-16e^{4x}$

Answers: $1) 2x+2;\ \ \ 2) \ 9xe^{3x}+6e^{3x}-16e^{4x}.$