Question

if (3x^{2}+5x + xy=3) and (y(3)= - 13), find (y(3)) by implicit differentiation.(y(3)=)

Explanation:

Step1: Differentiate each term

Differentiate $3x^{2}+5x + xy=3$ with respect to $x$.
The derivative of $3x^{2}$ is $6x$ (using power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$), the derivative of $5x$ is $5$, and for $xy$ we use the product rule $\frac{d}{dx}(uv)=u'v + uv'$ where $u = x$ and $v = y$. So $\frac{d}{dx}(xy)=y+xy'$. The derivative of the constant $3$ is $0$.
We get $6x + 5+y+xy'=0$.

Step2: Solve for $y'$

Rearrange the equation $6x + 5+y+xy'=0$ to isolate $y'$.
$xy'=-6x - 5 - y$.
$y'=\frac{-6x - 5 - y}{x}$.

Step3: Substitute $x = 3$ and $y(3)=-13$

Substitute $x = 3$ and $y=-13$ into the formula for $y'$.
$y'(3)=\frac{-6\times3 - 5-(-13)}{3}=\frac{-18 - 5 + 13}{3}=\frac{-10}{3}$.

Answer:

$-\frac{10}{3}$