if (4x^{2}+5x + xy=2) and (y(2)=-12), find (y(2)) by implicit differentiation.\n(y(2)=)

Explanation:

Step1: Differentiate both sides

Differentiate $4x^{2}+5x + xy=2$ with respect to $x$. The derivative of $4x^{2}$ is $8x$ (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 $2$ is $0$. We get $8x + 5+y+xy'=0$.

Step2: Solve for $y'$

Isolate $y'$: [ \begin{align*} xy'&=-8x - 5 - y\ y'&=\frac{-8x - 5 - y}{x} \end{align*} ]

Step3: Substitute $x = 2$ and $y(2)=-12$

Substitute $x = 2$ and $y=-12$ into the equation for $y'$: [ \begin{align*} y'(2)&=\frac{-8\times2-5-(-12)}{2}\ &=\frac{-16 - 5 + 12}{2}\ &=\frac{-16+7}{2}\ &=-\frac{9}{2} \end{align*} ]

Answer:

$-\frac{9}{2}$