implicit differentiation: problem 2\n(1 point)\nfind \\(\\frac{dy}{dx}\\) by implicit differentiation.\n\\(4…

Explanation:

Step1: Differentiate both sides

Differentiate $4 + 3x=\sin(xy^{2})$ with respect to $x$. The derivative of the left - hand side: $\frac{d}{dx}(4 + 3x)=\frac{d}{dx}(4)+\frac{d}{dx}(3x)=3$. For the right - hand side, use the chain rule. Let $u = xy^{2}$, then $\frac{d}{dx}\sin(xy^{2})=\cos(xy^{2})\cdot\frac{d}{dx}(xy^{2})$.

Step2: Differentiate $xy^{2}$ using product and chain rules

By the product rule $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$, where $u = x$ and $v = y^{2}$. So $\frac{d}{dx}(xy^{2})=y^{2}+x\cdot2y\frac{dy}{dx}$.

Step3: Set up the equation

We have $3=\cos(xy^{2})\left(y^{2}+2xy\frac{dy}{dx}\right)$.

Step4: Expand and solve for $\frac{dy}{dx}$

Expand the right - hand side: $3 = y^{2}\cos(xy^{2})+2xy\cos(xy^{2})\frac{dy}{dx}$. Then, move the term with $\frac{dy}{dx}$ to one side: $2xy\cos(xy^{2})\frac{dy}{dx}=3 - y^{2}\cos(xy^{2})$. Finally, $\frac{dy}{dx}=\frac{3 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$.

Answer:

$\frac{3 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$