Question

use implicit differentiation to determine $\frac{dy}{dx}$ given the equation $x^{2}+y^{6}=-9$.
$\frac{dy}{dx}=$

Explanation:

Step1: Differentiate both sides

Differentiate $x^{2}+y^{6}=-9$ with respect to $x$. The derivative of $x^{2}$ with respect to $x$ is $2x$ by the power - rule. For $y^{6}$, we use the chain - rule. Let $u = y$, then $\frac{d}{dx}(y^{6})=6y^{5}\frac{dy}{dx}$. The derivative of a constant $- 9$ with respect to $x$ is $0$. So we have $2x + 6y^{5}\frac{dy}{dx}=0$.

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

Subtract $2x$ from both sides: $6y^{5}\frac{dy}{dx}=-2x$. Then divide both sides by $6y^{5}$ (assuming $y
eq0$). We get $\frac{dy}{dx}=-\frac{2x}{6y^{5}}=-\frac{x}{3y^{5}}$.

Answer:

$-\frac{x}{3y^{5}}$