find the indicated derivative and simplify.\ny for y = (8 + 7x - 9x^2)e^x\ny =

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y'=u'v + uv'$. Let $u = 8 + 7x-9x^{2}$ and $v = e^{x}$. First, find $u'$ and $v'$. $u'=\frac{d}{dx}(8 + 7x-9x^{2})=7 - 18x$ and $v'=\frac{d}{dx}(e^{x})=e^{x}$.

Step2: Substitute into product - rule

$y'=(7 - 18x)e^{x}+(8 + 7x-9x^{2})e^{x}$.

Step3: Factor out $e^{x}$

$y'=e^{x}[(7 - 18x)+(8 + 7x-9x^{2})]$.

Step4: Simplify the expression inside the brackets

$y'=e^{x}(7 - 18x+8 + 7x-9x^{2})=e^{x}(-9x^{2}-11x + 15)$.

Answer:

$e^{x}(-9x^{2}-11x + 15)$