Question

use implicit differentiation to find \\( \frac { d y } { d x } \\) without first solving for \\( y \\).
\\( x ^ { 6 } + y ^ { 6 } = - 9 \\)
\\( \frac { d y } { d x } = \\)

Explanation:

Step1: Differentiate both sides with respect to \(x\)

Differentiate \(x^{6}+y^{6}=-9\).
Using the power rule \(\frac{d}{dx}(x^{n}) = nx^{n - 1}\) and the chain - rule \(\frac{d}{dx}(y^{n})=ny^{n - 1}\frac{dy}{dx}\).
We get \(\frac{d}{dx}(x^{6})+\frac{d}{dx}(y^{6})=\frac{d}{dx}(-9)\).
\(6x^{5}+6y^{5}\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

Subtract \(6x^{5}\) from both sides: \(6y^{5}\frac{dy}{dx}=-6x^{5}\).
Divide both sides by \(6y^{5}\): \(\frac{dy}{dx}=\frac{-6x^{5}}{6y^{5}}\).

Answer:

\(\frac{dy}{dx}=-\frac{x^{5}}{y^{5}}\)