differentiate the function.\ny=(x + 6)(x^3+2x + 1)\ny=

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u=x + 6$ and $v=x^{3}+2x + 1$.

Step2: Differentiate $u$

Differentiating $u=x + 6$ with respect to $x$, we get $u'=\frac{d}{dx}(x + 6)=1$.

Step3: Differentiate $v$

Differentiating $v=x^{3}+2x + 1$ with respect to $x$, we have $v'=\frac{d}{dx}(x^{3}+2x + 1)=3x^{2}+2$.

Step4: Substitute into product - rule

$y'=u'v+uv'=1\cdot(x^{3}+2x + 1)+(x + 6)\cdot(3x^{2}+2)$.

Step5: Expand the expression

$y'=x^{3}+2x + 1+3x^{3}+2x+18x^{2}+12$.

Step6: Combine like - terms

$y'=(x^{3}+3x^{3})+18x^{2}+(2x + 2x)+(1 + 12)=4x^{3}+18x^{2}+4x + 13$.

Answer:

$4x^{3}+18x^{2}+4x + 13$