Question

find $\frac{dy}{dx}$ by implicit differentiation.
$6 + 5x=sin(xy^{2})$
answer: $\frac{dy}{dx}=$

Explanation:

Step1: Differentiate both sides

Differentiate $6 + 5x$ and $\sin(xy^{2})$ with respect to $x$.
The derivative of $6+5x$ with respect to $x$ is $5$. For $\sin(xy^{2})$, by the chain - rule, let $u = xy^{2}$, then $\frac{d}{dx}\sin(u)=\cos(u)\cdot\frac{du}{dx}$. And $\frac{du}{dx}=y^{2}+2xy\frac{dy}{dx}$ (using product - rule for $xy^{2}$). So $\frac{d}{dx}\sin(xy^{2})=\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$.
So we have $5=\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$.

Step2: Expand the right - hand side

Expand $\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$ to get $y^{2}\cos(xy^{2})+2xy\cos(xy^{2})\frac{dy}{dx}$.
The equation becomes $5 = y^{2}\cos(xy^{2})+2xy\cos(xy^{2})\frac{dy}{dx}$.

Step3: Isolate $\frac{dy}{dx}$

First, move the term without $\frac{dy}{dx}$ to the left - hand side: $5 - y^{2}\cos(xy^{2})=2xy\cos(xy^{2})\frac{dy}{dx}$.
Then divide both sides by $2xy\cos(xy^{2})$ to solve for $\frac{dy}{dx}$.
$\frac{dy}{dx}=\frac{5 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$.

Answer:

$\frac{5 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$