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

Explanation:

Step1: Differentiate both sides

Differentiate $5y^{2}$ with respect to $x$ using the chain - rule ($\frac{d}{dx}(5y^{2}) = 10y\frac{dy}{dx}$), and differentiate $\frac{3x - 4}{3x+4}$ with respect to $x$ using the quotient - rule. The quotient - rule states that if $u = 3x - 4$ and $v = 3x + 4$, then $\frac{d}{dx}(\frac{u}{v})=\frac{u'v - uv'}{v^{2}}$, where $u'=3$ and $v' = 3$. So $\frac{d}{dx}(\frac{3x - 4}{3x + 4})=\frac{3(3x + 4)-3(3x - 4)}{(3x + 4)^{2}}=\frac{9x+12-(9x - 12)}{(3x + 4)^{2}}=\frac{24}{(3x + 4)^{2}}$. So we have $10y\frac{dy}{dx}=\frac{24}{(3x + 4)^{2}}$.

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

Divide both sides of the equation $10y\frac{dy}{dx}=\frac{24}{(3x + 4)^{2}}$ by $10y$ (assuming $y\neq0$). $\frac{dy}{dx}=\frac{24}{10y(3x + 4)^{2}}=\frac{12}{5y(3x + 4)^{2}}$.

Answer:

$\frac{12}{5y(3x + 4)^{2}}$