Question

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

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

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

Explanation:

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

Differentiate \(y^{4}\) using the chain - rule. The derivative of \(u^{n}\) with respect to \(x\) is \(n\cdot u^{n - 1}\cdot\frac{du}{dx}\), where \(u = y\) and \(n = 4\). The derivative of \(x^{9}\) with respect to \(x\) is \(9x^{8}\) using the power - rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\).
So, \(\frac{d}{dx}(y^{4})=\frac{d}{dx}(x^{9})\) gives \(4y^{3}\frac{dy}{dx}=9x^{8}\).

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

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

Answer:

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