find the derivative with respect to x if f(x) = (9x^2 - 18x + 18)e^x\nthe derivative with respect to x if…

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = 9x^{2}-18x + 18$ and $v = e^{x}$.

Step2: Find $u^\prime$

Differentiate $u = 9x^{2}-18x + 18$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u^\prime=\frac{d}{dx}(9x^{2}-18x + 18)=18x-18$.

Step3: Find $v^\prime$

Differentiate $v = e^{x}$ with respect to $x$. The derivative of $e^{x}$ is $e^{x}$, so $v^\prime=e^{x}$.

Step4: Calculate $f^\prime(x)$

Substitute $u$, $u^\prime$, $v$, and $v^\prime$ into the product - rule formula: [ \begin{align*} f^\prime(x)&=(18x - 18)e^{x}+(9x^{2}-18x + 18)e^{x}\ &=e^{x}(18x-18 + 9x^{2}-18x + 18)\ &=9x^{2}e^{x} \end{align*} ]

Answer:

$9x^{2}e^{x}$