use implicit differentiation to find \\( \\frac { d y } { d x } \\) without first solving for \\( y…

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

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

Answer

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}})