Question

use implicit differentiation to find $\frac{dy}{dx}$. $4y^{2}=\frac{5x - 2}{5x + 2}$ $\frac{dy}{dx}=\square$

Explanation:

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

Differentiate the left - hand side using the chain rule. If \(u = 4y^{2}\), then \(\frac{du}{dx}=4\times2y\frac{dy}{dx}=8y\frac{dy}{dx}\).

Differentiate the right - hand side using the quotient rule. If \(u = 5x - 2\) and \(v=5x + 2\), then \(\frac{d}{dx}(\frac{u}{v})=\frac{u'v - uv'}{v^{2}}\). Here, \(u' = 5\) and \(v'=5\). So \(\frac{d}{dx}(\frac{5x - 2}{5x + 2})=\frac{5(5x + 2)-5(5x - 2)}{(5x + 2)^{2}}=\frac{25x+10-(25x - 10)}{(5x + 2)^{2}}=\frac{20}{(5x + 2)^{2}}\).

So we have the equation \(8y\frac{dy}{dx}=\frac{20}{(5x + 2)^{2}}\).

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

Divide both sides of the equation \(8y\frac{dy}{dx}=\frac{20}{(5x + 2)^{2}}\) by \(8y\) (assuming \(y
eq0\)).

\(\frac{dy}{dx}=\frac{20}{8y(5x + 2)^{2}}=\frac{5}{2y(5x + 2)^{2}}\)

Answer:

\(\frac{5}{2y(5x + 2)^{2}}\)