use implicit differentiation to find $\frac{dy}{dx}$ and then $\frac{d^{2}y}{dx^{2}}$.\n$x^{\frac{2}{3}}+y^{\…

use implicit differentiation to find $\frac{dy}{dx}$ and then $\frac{d^{2}y}{dx^{2}}$.\n$x^{\frac{2}{3}}+y^{\frac{2}{3}} = 126$\n$\frac{dy}{dx}=square$ (type an exact answer.)

use implicit differentiation to find $\frac{dy}{dx}$ and then $\frac{d^{2}y}{dx^{2}}$.\n$x^{\frac{2}{3}}+y^{\frac{2}{3}} = 126$\n$\frac{dy}{dx}=square$ (type an exact answer.)

Answer

Explanation:

Step1: Differentiate both sides with respect to x

Differentiate $x^{\frac{2}{3}}+y^{\frac{2}{3}} = 126$ term - by - term. The derivative of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{2}{3}x^{-\frac{1}{3}}$ by the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $y^{\frac{2}{3}}$, using the chain - rule $\frac{d}{dx}(y^{\frac{2}{3}})=\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}$. The derivative of the constant 126 with respect to $x$ is 0. So we have $\frac{2}{3}x^{-\frac{1}{3}}+\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}=0$.

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

First, subtract $\frac{2}{3}x^{-\frac{1}{3}}$ from both sides: $\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}=-\frac{2}{3}x^{-\frac{1}{3}}$. Then divide both sides by $\frac{2}{3}y^{-\frac{1}{3}}$ to get $\frac{dy}{dx}=-\left(\frac{y}{x}\right)^{\frac{1}{3}}$.

Answer:

$-\left(\frac{y}{x}\right)^{\frac{1}{3}}$