Question

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

\\( y ^ { 9 } = x ^ { 4 } \\)

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

Explanation:

Step1: Differentiate both sides

Differentiate \(y^{9}\) and \(x^{4}\) with respect to \(x\).
Using the chain rule \((u^n)^\prime = nu^{n - 1}u^\prime\) for \(y^{9}\) (where \(u = y\)) and the power rule \((x^n)^\prime=nx^{n - 1}\) for \(x^{4}\).
We get \(9y^{8}\frac{dy}{dx}=4x^{3}\).

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

Divide both sides of the equation \(9y^{8}\frac{dy}{dx}=4x^{3}\) by \(9y^{8}\).
So \(\frac{dy}{dx}=\frac{4x^{3}}{9y^{8}}\).

Answer:

\(\frac{4x^{3}}{9y^{8}}\)