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

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

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

Answer

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\neq0)).

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

Answer:

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