find g(x) if g(x) = 9xe^6x. g(x)=

Explanation:

Step1: Identify product - rule

The function $g(x)=9xe^{6x}$ is a product of $u = 9x$ and $v=e^{6x}$. The product - rule states that $(uv)'=u'v + uv'$.

Step2: Find $u'$

If $u = 9x$, then by the power - rule, $u'=\frac{d}{dx}(9x)=9$.

Step3: Find $v'$

If $v = e^{6x}$, then by the chain - rule. Let $t = 6x$, $\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}$. Since $\frac{d}{dt}(e^{t})=e^{t}$ and $\frac{dt}{dx}=6$, $v'=\frac{d}{dx}(e^{6x})=e^{6x}\cdot6 = 6e^{6x}$.

Step4: Apply product - rule

$g'(x)=u'v+uv'$. Substitute $u = 9x$, $u' = 9$, $v = e^{6x}$, and $v' = 6e^{6x}$ into the formula: $g'(x)=9\cdot e^{6x}+9x\cdot6e^{6x}$.

Step5: Simplify the result

$g'(x)=9e^{6x}(1 + 6x)$.

Answer:

$9e^{6x}(1 + 6x)$