use implicit differentiation to determine \\( \\frac { d y } { d x } \\) given the equation \\( x ^ { 6 } +…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x^{6}+y^{5}=7) term - by - term. 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^{5})=\frac{d}{dx}(7)). (6x^{5}+5y^{4}\frac{dy}{dx}=0).

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

Subtract (6x^{5}) from both sides: (5y^{4}\frac{dy}{dx}=-6x^{5}). Then divide both sides by (5y^{4}) (assuming (y\neq0)). (\frac{dy}{dx}=-\frac{6x^{5}}{5y^{4}}).

Answer:

(-\frac{6x^{5}}{5y^{4}})