differentiate implicitly to find $\frac{dy}{dx}$. 6xy + 7 = 0 $\frac{dy}{dx}=square$

Explanation:

Step1: Differentiate each term

Differentiate $6xy$ using product - rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = 6x$ and $v = y$. The derivative of $6x$ with respect to $x$ is $6$, and the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$. The derivative of the constant $7$ with respect to $x$ is $0$. So, $\frac{d}{dx}(6xy)+\frac{d}{dx}(7)=\frac{d}{dx}(0)$. $6y + 6x\frac{dy}{dx}+0 = 0$

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

Subtract $6y$ from both sides: $6x\frac{dy}{dx}=-6y$. Then divide both sides by $6x$ ($x\neq0$): $\frac{dy}{dx}=-\frac{y}{x}$

Answer:

$-\frac{y}{x}$