Question

find y. y = x(2x + 1)^4

Explanation:

Step1: Apply product rule for first - derivative

The product rule is $(uv)' = u'v+uv'$, where $u = x$ and $v=(2x + 1)^4$.
$u'=1$, and by the chain - rule, if $y=(2x + 1)^4$, let $t = 2x+1$, then $y=t^4$, and $\frac{dy}{dt}=4t^3$, $\frac{dt}{dx}=2$, so $\frac{dy}{dx}=8(2x + 1)^3$.
$y'=1\times(2x + 1)^4+x\times8(2x + 1)^3=(2x + 1)^3(2x + 1+8x)=(2x + 1)^3(10x + 1)$.

Step2: Apply product rule for second - derivative

Let $u=(2x + 1)^3$ and $v = 10x+1$.
By the chain - rule, $u'=3\times2\times(2x + 1)^2=6(2x + 1)^2$, and $v' = 10$.
$y''=u'v+uv'=6(2x + 1)^2(10x + 1)+(2x + 1)^3\times10$.
Factor out $(2x + 1)^2$:
$y''=(2x + 1)^2[6(10x + 1)+10(2x + 1)]=(2x + 1)^2(60x+6 + 20x+10)=(2x + 1)^2(80x + 16)=16(2x + 1)^2(5x + 1)$.

Answer:

$16(2x + 1)^2(5x + 1)$