Question

let $y = \frac{14cdot x + 10}{12cdot x^{2}+9cdot x + 7}$. find $\frac{dy}{dx}$. $\frac{dy}{dx}=$
use $(3x + 5)^2$ to square $(3x + 5)$. use \/\ to divide. do not simplify your answer.

Explanation:

Step1: Identify the quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{u'v - uv'}{v^{2}}$, where $u = 14x + 10$ and $v=12x^{2}+9x + 7$.

Step2: Find the derivative of $u$

Differentiate $u = 14x+10$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax + b)=a$, we get $u'=\frac{d}{dx}(14x + 10)=14$.

Step3: Find the derivative of $v$

Differentiate $v = 12x^{2}+9x + 7$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $v'=\frac{d}{dx}(12x^{2}+9x + 7)=24x+9$.

Step4: Apply the quotient - rule

Substitute $u$, $u'$, $v$, and $v'$ into the quotient - rule formula:
$\frac{dy}{dx}=\frac{14(12x^{2}+9x + 7)-(14x + 10)(24x + 9)}{(12x^{2}+9x + 7)^{2}}$

Answer:

$\frac{14(12x^{2}+9x + 7)-(14x + 10)(24x + 9)}{(12x^{2}+9x + 7)^{2}}$