10. -/2 points details my notes find f (x) and f (x). f(x)=x^6e^x f (x)= f (x)= need help? read it watch it…

Explanation:

Step1: Apply product - rule for first - derivative

The product rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = x^{6}$ and $v = e^{x}$. The derivative of $u=x^{6}$ is $u' = 6x^{5}$ and the derivative of $v = e^{x}$ is $v'=e^{x}$. So, $f'(x)=(x^{6})'e^{x}+x^{6}(e^{x})'=6x^{5}e^{x}+x^{6}e^{x}=x^{5}e^{x}(6 + x)$.

Step2: Apply product - rule for second - derivative

Now, to find $f''(x)$, we consider $f'(x)=x^{5}e^{x}(6 + x)=(6x^{5}+x^{6})e^{x}$. Let $u = 6x^{5}+x^{6}$ and $v = e^{x}$. Then $u'=30x^{4}+6x^{5}$ and $v' = e^{x}$. So, $f''(x)=(30x^{4}+6x^{5})e^{x}+(6x^{5}+x^{6})e^{x}=e^{x}(30x^{4}+12x^{5}+x^{6})$.

Answer:

$f'(x)=x^{5}e^{x}(x + 6)$ $f''(x)=e^{x}(x^{6}+12x^{5}+30x^{4})$