Question

differentiate implicitly to find \\( \frac { d y } { d x } \\)

\\( y ^ { 8 } = x ^ { 7 } \\)

\\( \frac { d y } { d x } = \\)

Explanation:

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

Differentiate \(y^{8}\) using the chain rule \(\frac{d}{dx}(y^{8}) = 8y^{7}\frac{dy}{dx}\), and differentiate \(x^{7}\) using the power rule \(\frac{d}{dx}(x^{7})=7x^{6}\). So we have \(8y^{7}\frac{dy}{dx}=7x^{6}\).

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

Divide both sides of the equation \(8y^{7}\frac{dy}{dx}=7x^{6}\) by \(8y^{7}\). We get \(\frac{dy}{dx}=\frac{7x^{6}}{8y^{7}}\).

Answer:

\(\frac{7x^{6}}{8y^{7}}\)