find dy/dx by implicit differentiation. 1 + 6x = sin(xy²) dy/dx =

Explanation:

Step1: Differentiate both sides

Differentiate $1 + 6x$ and $\sin(xy^{2})$ with respect to $x$. The derivative of $1+6x$ with respect to $x$ is $6$. For $\sin(xy^{2})$, use the chain - rule. Let $u = xy^{2}$, then $\frac{d}{dx}\sin(u)=\cos(u)\cdot\frac{du}{dx}$. Now find $\frac{du}{dx}$ using the product - rule. $\frac{du}{dx}=y^{2}+2xy\frac{dy}{dx}$. So $\frac{d}{dx}\sin(xy^{2})=\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$. We get $6=\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$.

Step2: Expand the right - hand side

Expand $\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$ to get $y^{2}\cos(xy^{2})+2xy\cos(xy^{2})\frac{dy}{dx}$. So the equation is $6 = y^{2}\cos(xy^{2})+2xy\cos(xy^{2})\frac{dy}{dx}$.

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

First, move the term without $\frac{dy}{dx}$ to the left - hand side: $6 - y^{2}\cos(xy^{2})=2xy\cos(xy^{2})\frac{dy}{dx}$. Then divide both sides by $2xy\cos(xy^{2})$ to solve for $\frac{dy}{dx}$. $\frac{dy}{dx}=\frac{6 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$

Answer:

$\frac{6 - y^{2}\cos(xy^{2})}{2xy\cos(xy^{2})}$