Question

find the derivative of the function.
$y = x^{2}e^{-4x}$
$y=square$
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Explanation:

Step1: Apply product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y'=u'v + uv'$. Here, $u = x^{2}$ and $v=e^{-4x}$.

Step2: Find $u'$

Differentiate $u = x^{2}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $u'=\frac{d}{dx}(x^{2}) = 2x$.

Step3: Find $v'$

Differentiate $v = e^{-4x}$ with respect to $x$. Using the chain - rule $\frac{d}{dx}(e^{ax})=ae^{ax}$, we get $v'=\frac{d}{dx}(e^{-4x})=-4e^{-4x}$.

Step4: Calculate $y'$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula:

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Answer:

$2xe^{-4x}(1 - 2x)$