Full solution

Q. Using implicit differentiation, find

\frac{d y}{d x}

\newline

\sqrt{x y}=-6+x y

Write Equation: First, let's write down the equation we need to differentiate implicitly: $\newline$ $\sqrt{xy} = -6 + xy$ $\newline$ We need to differentiate both sides of the equation with respect to $x$ .
Differentiate Left Side: Differentiate the left side of the equation with respect to $x$ : The left side is $\sqrt{xy}$ , which is $(xy)^{1/2}$ . Using the chain rule, we differentiate $(xy)^{1/2}$ with respect to $x$ . The derivative of $u^{1/2}$ with respect to $u$ is $(1/2)u^{-1/2}$ , and then we multiply by the derivative of $u$ with respect to $x$ , where $\sqrt{xy}$ $0$ . So we get $\sqrt{xy}$ $1$ .
Differentiate Right Side: Differentiate the right side of the equation with respect to $x$ : The right side is $-6 + xy$ . The derivative of a constant is $0$ , so the derivative of $-6$ is $0$ . The derivative of $xy$ with respect to $x$ is $y + x\frac{dy}{dx}$ because we apply the product rule.
Equate Derivatives: Now we equate the derivatives from the left and right sides: $\newline$ $(\frac{1}{2})(xy)^{-\frac{1}{2}} * (y + x\frac{dy}{dx}) = 0 + y + x\frac{dy}{dx}$
Simplify and Solve: Simplify the equation and solve for $\frac{dy}{dx}$ : We multiply both sides by $2(xy)^{\frac{1}{2}}$ to get rid of the fraction and the square root: $y + x\frac{dy}{dx} = 2(xy)^{\frac{1}{2}}(y + x\frac{dy}{dx})$
Distribute Right Side: Distribute the right side of the equation: $\newline$ $y + x\frac{dy}{dx} = 2y(xy)^{\frac{1}{2}} + 2x^2\frac{dy}{dx}(xy)^{\frac{1}{2}}$
Group Terms: Group the terms with $\frac{dy}{dx}$ on one side and the other terms on the opposite side: $\newline$ $x\frac{dy}{dx} - 2x^2\frac{dy}{dx}(xy)^{\frac{1}{2}} = 2y(xy)^{\frac{1}{2}} - y$
Factor Out: Factor out $(\frac{dy}{dx})$ from the terms on the left side: $\newline$ (\frac{dy}{dx})(x - \(2x^ $2$ (xy)^{\frac{ $1$ }{ $2$ }}) = $2$ y(xy)^{\frac{ $1$ }{ $2$ }} - y
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{2y(xy)^{\frac{1}{2}} - y}{x - 2x^2(xy)^{\frac{1}{2}}}$
Simplify Expression: We can simplify the expression further by factoring out $y$ from the numerator: $\frac{dy}{dx} = \frac{y(2(xy)^{\frac{1}{2}} - 1)}{(x - 2x^2(xy)^{\frac{1}{2}})}$