use implicit differentiation to find $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$.\n$2x^{2}+y^{2}=1$\n$\frac{d…

use implicit differentiation to find $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$.\n$2x^{2}+y^{2}=1$\n$\frac{dy}{dx}=square$

use implicit differentiation to find $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$.\n$2x^{2}+y^{2}=1$\n$\frac{dy}{dx}=square$

Answer

Explanation:

Step1: Differentiate both sides with respect to x

Differentiate $2x^{2}+y^{2}=1$ term - by - term. The derivative of $2x^{2}$ with respect to $x$ is $4x$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$. For $y^{2}$, by the chain - rule $\frac{d}{dx}(y^{2}) = 2y\frac{dy}{dx}$. The derivative of the constant 1 is 0. So we have $4x + 2y\frac{dy}{dx}=0$.

Step2: Solve for $\frac{dy}{dx}$

Subtract $4x$ from both sides: $2y\frac{dy}{dx}=-4x$. Then divide both sides by $2y$ (assuming $y\neq0$) to get $\frac{dy}{dx}=-\frac{2x}{y}$.

Answer:

$-\frac{2x}{y}$