differentiate the function. y=(x + 4)(x²+4x+3) y=□

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u=(x + 4)$ and $v=(x^{2}+4x + 3)$.

Step2: Find $u'$

Differentiate $u=(x + 4)$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $u'=\frac{d}{dx}(x+4)=1$.

Step3: Find $v'$

Differentiate $v=(x^{2}+4x + 3)$ with respect to $x$. Using the power - rule, $v'=\frac{d}{dx}(x^{2}+4x + 3)=2x+4$.

Step4: Calculate $y'$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula: $y'=u'v+uv'=1\cdot(x^{2}+4x + 3)+(x + 4)\cdot(2x + 4)$. Expand the expression: [ \begin{align*} y'&=x^{2}+4x + 3+(x(2x + 4)+4(2x + 4))\ &=x^{2}+4x + 3+(2x^{2}+4x+8x + 16)\ &=x^{2}+4x + 3+2x^{2}+12x + 16\ &=3x^{2}+16x + 19 \end{align*} ]

Answer:

$3x^{2}+16x + 19$