use the product rule to find the derivative.\ny=(5x^2 + 3)(2x - 5)\ny=□

Explanation:

Step1: Recall product - rule

The product rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u = 5x^{2}+3$ and $v = 2x - 5$.

Step2: Find $u'$

Differentiate $u = 5x^{2}+3$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u'=\frac{d}{dx}(5x^{2}+3)=10x$.

Step3: Find $v'$

Differentiate $v = 2x - 5$ with respect to $x$. Using the power - rule, $v'=\frac{d}{dx}(2x - 5)=2$.

Step4: Apply product - rule

Substitute $u$, $u'$, $v$, and $v'$ into the product rule formula: [ \begin{align*} y'&=u'v+uv'\ &=(10x)(2x - 5)+(5x^{2}+3)\times2\ &=20x^{2}-50x + 10x^{2}+6\ &=30x^{2}-50x + 6 \end{align*} ]

Answer:

$30x^{2}-50x + 6$