Question

use the product rule to find the derivative. y=(2x^{2}+5)(5x - 3) y=□

Explanation:

Step1: Recall product - rule

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

Step2: Find $u'$

Differentiate $u = 2x^{2}+5$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u'=\frac{d}{dx}(2x^{2}+5)=4x$.

Step3: Find $v'$

Differentiate $v = 5x - 3$ with respect to $x$. Using the power - rule, $v'=\frac{d}{dx}(5x - 3)=5$.

Step4: Apply product rule

Substitute $u$, $u'$, $v$, and $v'$ into the product rule formula $y'=u'v+uv'$.
$y'=(4x)(5x - 3)+(2x^{2}+5)\times5$.
Expand the expression:
\[

$$\begin{align*} y'&=20x^{2}-12x + 10x^{2}+25\\ &=(20x^{2}+10x^{2})-12x + 25\\ &=30x^{2}-12x + 25 \end{align*}$$

\]

Answer:

$30x^{2}-12x + 25$