Question

if (3x^{3}+5x^{2}y^{2}=2y), then (\frac{dy}{dx}=) a (\frac{-6x}{20xy - 2}) b (\frac{-10xy^{2}-6x}{10x^{2}y - 2}) c (\frac{3 - 6x-10xy^{2}}{10x^{2}y}) d (6xcdot\frac{10x^{2}y + 10xy^{2}}{1})

Explanation:

Step1: Differentiate implicitly

Differentiate each term of $3x^{2}+5x^{2}y^{2}=2y$ with respect to $x$.
For $3x^{2}$, using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $\frac{d}{dx}(3x^{2}) = 6x$.
For $5x^{2}y^{2}$, use the product - rule $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$ where $u = 5x^{2}$ and $v=y^{2}$.
$\frac{d}{dx}(5x^{2}y^{2})=5x^{2}\cdot2y\frac{dy}{dx}+5y^{2}\cdot2x=10x^{2}y\frac{dy}{dx}+10xy^{2}$.
For $2y$, we have $\frac{d}{dx}(2y)=2\frac{dy}{dx}$.
So, $6x + 10x^{2}y\frac{dy}{dx}+10xy^{2}=2\frac{dy}{dx}$.

Step2: Isolate $\frac{dy}{dx}$

Move all terms with $\frac{dy}{dx}$ to one side:
$10x^{2}y\frac{dy}{dx}-2\frac{dy}{dx}=-6x - 10xy^{2}$.
Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(10x^{2}y - 2)=-6x - 10xy^{2}$.
Then $\frac{dy}{dx}=\frac{-6x - 10xy^{2}}{10x^{2}y - 2}$.

Answer:

B. $\frac{-10xy^{2}-6x}{10x^{2}y - 2}$