Question

use implicit differentiation to determine \\(\frac{dy}{dx}\\) for the equation \\(\frac{2}{x} - \frac{3}{y} = 7\\).

Explanation:

Rewrite the equation using negative exponents

$$ 2x^{-1} - 3y^{-1} = 7 $$

Differentiate both sides with respect to \(x\)

$$ -2x^{-2} - 3(-1)y^{-2}\frac{dy}{dx} = 0 $$

$$ -\frac{2}{x^2} + \frac{3}{y^2}\frac{dy}{dx} = 0 $$

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

$$ \frac{3}{y^2}\frac{dy}{dx} = \frac{2}{x^2} $$

$$ \frac{dy}{dx} = \frac{2y^2}{3x^2} $$

Answer:

Use implicit differentiation to determine \(\frac{dy}{dx}\) for the equation \(\frac{2}{x} - \frac{3}{y} = 7\).

\(\frac{dy}{dx} =\) <blank>\(\frac{2y^2}{3x^2}\)</blank>