use implicit differentiation to find $\frac{dy}{dx}$. $5y^{2}=\frac{4x - 3}{4x + 3}$ $\frac{dy}{dx}=square$
use implicit differentiation to find $\frac{dy}{dx}$. $5y^{2}=\frac{4x - 3}{4x + 3}$ $\frac{dy}{dx}=square$
Answer
Explanation:
Step1: Differentiate both sides
Differentiate $5y^{2}$ and $\frac{4x - 3}{4x+3}$ with respect to $x$. For the left - hand side, using the chain rule, $\frac{d}{dx}(5y^{2})=10y\frac{dy}{dx}$. For the right - hand side, use the quotient rule $\left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}}$, where $u = 4x-3$, $u'=4$, $v = 4x + 3$, $v'=4$. So $\frac{d}{dx}\left(\frac{4x - 3}{4x+3}\right)=\frac{4(4x + 3)-4(4x - 3)}{(4x + 3)^{2}}=\frac{16x+12-(16x - 12)}{(4x + 3)^{2}}=\frac{24}{(4x + 3)^{2}}$. So we have $10y\frac{dy}{dx}=\frac{24}{(4x + 3)^{2}}$.
Step2: Solve for $\frac{dy}{dx}$
Divide both sides of the equation $10y\frac{dy}{dx}=\frac{24}{(4x + 3)^{2}}$ by $10y$ (assuming $y\neq0$). $\frac{dy}{dx}=\frac{24}{10y(4x + 3)^{2}}=\frac{12}{5y(4x + 3)^{2}}$.
Answer:
$\frac{12}{5y(4x + 3)^{2}}$