Question

use implicit differentiation to find \\( \frac { d y } { d x } \\).
\\( 6 x ^ { 2 } y + 7 x y ^ { 2 } = - 6 \\)
\\( \frac { d y } { d x } = \\)

Explanation:

Step1: Differentiate both sides

Differentiate \(6x^{2}y + 7xy^{2}=-6\) with respect to \(x\) using the product rule \((uv)^\prime = u^\prime v+uv^\prime\).
For \(6x^{2}y\), let \(u = 6x^{2}\) and \(v=y\). Then \((6x^{2}y)^\prime=12xy + 6x^{2}\frac{dy}{dx}\).
For \(7xy^{2}\), let \(u = 7x\) and \(v = y^{2}\). Then \((7xy^{2})^\prime=7y^{2}+14xy\frac{dy}{dx}\).
The derivative of \(-6\) is \(0\). So \(12xy + 6x^{2}\frac{dy}{dx}+7y^{2}+14xy\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

Group the terms with \(\frac{dy}{dx}\): \((6x^{2}+14xy)\frac{dy}{dx}=-12xy - 7y^{2}\).
Factor out \(2x\) from \(6x^{2}+14xy\) (get \(2x(3x + 7y)\)) and factor out \(-y\) from \(-12xy-7y^{2}\) (get \(-y(12x + 7y)\)).
Then \(\frac{dy}{dx}=\frac{-12xy - 7y^{2}}{6x^{2}+14xy}=\frac{-y(12x + 7y)}{2x(3x + 7y)}\).

Answer:

\(\frac{-y(12x + 7y)}{2x(3x + 7y)}\)