use implicit differentiation to find dy/dx. 3y^2 = (4x - 5)/(4x + 5) dy/dx = □

Explanation:

Step1: Differentiate both sides

Differentiate $3y^{2}$ with respect to $x$ using chain - rule and $\frac{4x - 5}{4x+5}$ using quotient - rule. The derivative of $3y^{2}$ with respect to $x$ is $6y\frac{dy}{dx}$ (by chain - rule: $\frac{d}{dx}(3y^{2})=3\times2y\times\frac{dy}{dx}$). The quotient - rule states that if $u = 4x - 5$ and $v = 4x + 5$, then $\frac{d}{dx}(\frac{u}{v})=\frac{u'v - uv'}{v^{2}}$, where $u'=4$ and $v' = 4$. So, $\frac{d}{dx}(\frac{4x - 5}{4x + 5})=\frac{4(4x + 5)-4(4x - 5)}{(4x + 5)^{2}}=\frac{16x+20-(16x - 20)}{(4x + 5)^{2}}=\frac{40}{(4x + 5)^{2}}$. We get $6y\frac{dy}{dx}=\frac{40}{(4x + 5)^{2}}$.

Step2: Solve for $\frac{dy}{dx}$

Divide both sides of the equation $6y\frac{dy}{dx}=\frac{40}{(4x + 5)^{2}}$ by $6y$ (assuming $y\neq0$). $\frac{dy}{dx}=\frac{40}{6y(4x + 5)^{2}}=\frac{20}{3y(4x + 5)^{2}}$.

Answer:

$\frac{20}{3y(4x + 5)^{2}}$