Question

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

\\( e ^ { 5 x y } = y ^ { 8 } \\)

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

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Explanation:

Step1: Differentiate both sides

Differentiate \(e^{5xy}\) using the chain rule and product rule. The derivative of \(e^{u}\) with respect to \(x\) is \(e^{u}\cdot u'\), where \(u = 5xy\). So, \((e^{5xy})'=e^{5xy}\cdot(5y + 5x\frac{dy}{dx})\).

Differentiate \(y^{8}\) using the chain rule. The derivative is \(8y^{7}\frac{dy}{dx}\).

So we have \(e^{5xy}(5y + 5x\frac{dy}{dx})=8y^{7}\frac{dy}{dx}\).

Step2: Expand and collect \(\frac{dy}{dx}\) terms

Expand the left - hand side: \(5ye^{5xy}+5xe^{5xy}\frac{dy}{dx}=8y^{7}\frac{dy}{dx}\).

Move all terms with \(\frac{dy}{dx}\) to one side: \(5xe^{5xy}\frac{dy}{dx}-8y^{7}\frac{dy}{dx}=- 5ye^{5xy}\).

Factor out \(\frac{dy}{dx}\): \(\frac{dy}{dx}(5xe^{5xy}-8y^{7})=-5ye^{5xy}\).

Step3: Solve for \(\frac{dy}{dx}\)

\(\frac{dy}{dx}=\frac{-5ye^{5xy}}{5xe^{5xy}-8y^{7}}=\frac{5ye^{5xy}}{8y^{7}-5xe^{5xy}}\)

Answer:

\(\frac{5ye^{5xy}}{8y^{7}-5xe^{5xy}}\)