Question

use implicit differentiation to determine \\(\frac{dy}{dx}\\) for the equation \\(\frac{5}{x} - \frac{3}{y} = 5\\).

Explanation:

Rewrite the equation with negative exponents

$$ 5x^{-1} - 3y^{-1} = 5 $$

Differentiate both sides with respect to x

$$ -5x^{-2} - 3(-1)y^{-2}\frac{dy}{dx} = 0 $$

$$ -\frac{5}{x^2} + \frac{3}{y^2}\frac{dy}{dx} = 0 $$

Solve for dy/dx

$$ \frac{3}{y^2}\frac{dy}{dx} = \frac{5}{x^2} $$

$$ \frac{dy}{dx} = \frac{5y^2}{3x^2} $$

Answer:

Use implicit differentiation to determine \(\frac{dy}{dx}\) for the equation \(\frac{5}{x} - \frac{3}{y} = 5\).

\(\frac{dy}{dx} =\) <blank>\(\frac{5y^2}{3x^2}\)</blank>