Question

find g(x) if g(x) = 6xe^{8x}.
g(x)=□

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, $u = 6x$ and $v = e^{8x}$.

Step2: Find $u'$

Differentiate $u = 6x$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$, for $n = 1$ and $a = 6$, we have $u'=\frac{d}{dx}(6x)=6$.

Step3: Find $v'$

Differentiate $v = e^{8x}$ with respect to $x$. Using the chain - rule $\frac{d}{dx}(e^{ax})=ae^{ax}$, for $a = 8$, we have $v'=\frac{d}{dx}(e^{8x})=8e^{8x}$.

Step4: Calculate $g'(x)$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula $g'(x)=u'v+uv'$. So $g'(x)=6e^{8x}+6x\times8e^{8x}$.
Simplify to get $g'(x)=6e^{8x}(1 + 8x)$.

Answer:

$6e^{8x}(1 + 8x)$