Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = x ^ { 2 } ( 3 x + 2 ) ( 5 x + 2 )$
A) $65 x ^ { 2 } + ( 3 x + 48 ) ( 5 x + 2 )$
B) $60 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
C) $35 x ^ { 2 } + ( 3 x + 2 ) ( 5 x + 2 )$
D) $3 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
E) $3 x ^ { 2 } + 48 x + 8$

Question

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ .  $y = x ^ { 2 } ( 3 x + 2 ) ( 5 x + 2 )$  
A) $65 x ^ { 2 } + ( 3 x + 48 ) ( 5 x + 2 )$
B) $60 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
C) $35 x ^ { 2 } + ( 3 x + 2 ) ( 5 x + 2 )$
D) $3 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
E) $3 x ^ { 2 } + 48 x + 8$

Quizplus · Accepted Answer

To find the derivative of $y = x^2(3x + 2)(5x + 2)$, we apply the product rule. The product rule for three functions $f(x)g(x)h(x)$ is $f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)$. Here, $f(x) = x^2$, $g(x) = 3x + 2$, and $h(x) = 5x + 2$. The derivatives are $f'(x) = 2x$, $g'(x) = 3$, and $h'(x) = 5$. Substituting these into the product rule formula and simplifying gives $60x^3 + 48x^2 + 8x$.

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