Finding a Derivative Using Implicit Differentiation

Find \(\dfrac{dy}{dx}\) if \(3x^{2}+4y^{2}=2x\).

Solution We assume that \(y\) is a differentiable function of \(x\) and differentiate both sides with respect to \(x\). \[ \begin{eqnarray*} \begin{array}{rl@{\qquad}l} \dfrac{d}{dx} ( 3x^{2}+4y^{2}) &= \dfrac{d}{dx}(2x) & {\color{#0066A7}{\hbox{Differentiate both sides with respect to}\; x.}}\\ \dfrac{d}{dx} (3x^{2}) +\dfrac{d}{dx}( 4y^{2}) &= 2 & {\color{#0066A7}{\hbox{Sum Rule}.}} \\ 3\dfrac{d}{dx}x^{2}+4\dfrac{d}{dx}y^{2} &=2 &{\color{#0066A7}{\hbox{Constant Multiple Rule}.}}\\ 6x+4\!\left( 2y\dfrac{dy}{dx}\right) &= 2 &{\color{#0066A7}{\dfrac{d}{dx}{y}^{{ 2}}{=2y} \dfrac{dy}{dx}}}\\ 6x+8y\dfrac{dy}{dx} &= {2} & {\color{#0066A7}{\hbox{Simplify.}}}\\ \dfrac{dy}{dx} &= \dfrac{2-6x}{8y}=\dfrac{1-3x}{4y} & {\color{#0066A7}{\hbox{Solve for}\; \dfrac{dy}{dx}.}} \end{array} \end{eqnarray*} \] provided \(y\ne 0\).