question 1 (1 point) find the derivative of y=(2x + 1)^5(3x - 2)^7 and choose the correct option. dy/dx=(2x…

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, where $u=(2x + 1)^5$ and $v=(3x - 2)^7$, then $y^\prime=u^\prime v+uv^\prime$. First, find $u^\prime$ using the chain - rule. If $u=(2x + 1)^5$, let $t = 2x+1$, then $u = t^5$. By the chain - rule, $\frac{du}{dx}=\frac{du}{dt}\cdot\frac{dt}{dx}$. $\frac{du}{dt}=5t^4 = 5(2x + 1)^4$ and $\frac{dt}{dx}=2$, so $u^\prime=10(2x + 1)^4$. Second, find $v^\prime$ using the chain - rule. If $v=(3x - 2)^7$, let $s = 3x - 2$, then $v = s^7$. By the chain - rule, $\frac{dv}{dx}=\frac{dv}{ds}\cdot\frac{ds}{dx}$. $\frac{dv}{ds}=7s^6 = 7(3x - 2)^6$ and $\frac{ds}{dx}=3$, so $v^\prime=21(3x - 2)^6$.

Step2: Calculate $y^\prime$

$y^\prime=u^\prime v+uv^\prime=10(2x + 1)^4(3x - 2)^7+21(2x + 1)^5(3x - 2)^6$. Factor out the common factors $(2x + 1)^4(3x - 2)^6$: [ \begin{align*} y^\prime&=(2x + 1)^4(3x - 2)^6[10(3x - 2)+21(2x + 1)]\ &=(2x + 1)^4(3x - 2)^6(30x-20 + 42x+21)\ &=(2x + 1)^4(3x - 2)^6(72x + 1) \end{align*} ]

Answer:

$\frac{dy}{dx}=(2x + 1)^4(3x - 2)^6(72x + 1)$