Using implicit differentiation, find (dy)/(dx). sqrt(xy)=-5+yx^(3)

Apply Implicit Differentiation: First, we need to apply implicit differentiation to both sides of the equation with respect to x. The equation is sqrt(xy) = -5 + yx^3. We will differentiate term by term.Differentiate Left Side: Differentiate the left side, sqrt(xy), with respect to x. Using the chain rule, we get (1/2)(xy)^(-1/2) * (x(dy/dx) + y).Differentiate Right Side: Differentiate the right side, -5 + yx^3, with respect to x. The derivative of -5 is 0, and using the product rule for yx^3, we get 3yx^2 + x^3(dy/dx).Equate Derivatives: Now we equate the derivatives from the left and right sides to get (1/2)(xy)^(-1/2) * (x(dy/dx) + y) = 3yx^2 + x^3(dy/dx).Solve for (dy/dx): We need to solve for (dy/dx). To do this, we'll collect all the terms containing (dy/dx) on one side and the rest on the other side. This gives us (1/2)(xy)^(-1/2) * x(dy/dx) - x^3(dy/dx) = 3yx^2 - (1/2)(xy)^(-1/2) * y.Factor Out (dy/dx): Factor out (dy/dx) from the terms on the left side to get (dy/dx)((1/2)(xy)^(-1/2) * x - x^3) = 3yx^2 - (1/2)(xy)^(-1/2) * y.Isolate (dy/dx): Divide both sides by ((1/2)(xy)^(-1/2) * x - x^3) to isolate (dy/dx). This gives us (dy/dx) = (3yx^2 - (1/2)(xy)^(-1/2) * y) / ((1/2)(xy)^(-1/2) * x - x^3).Simplify (dy/dx): Simplify the expression for (dy/dx) if possible. However, in this case, the expression is already in its simplest form, so we have our final answer.

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Find derivatives using implicit differentiation

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Q. Using implicit differentiation, find

\frac{d y}{d x}

\newline

\sqrt{x y}=-5+y x^{3}

Apply Implicit Differentiation: First, we need to apply implicit differentiation to both sides of the equation with respect to $x$ . The equation is $\sqrt{xy} = -5 + yx^3$ . We will differentiate term by term.
Differentiate Left Side: Differentiate the left side, $\sqrt{xy}$ , with respect to $x$ . Using the chain rule, we get $\frac{1}{2}(xy)^{-\frac{1}{2}} \times (x\frac{dy}{dx} + y)$ .
Differentiate Right Side: Differentiate the right side, $-5 + yx^3$ , with respect to $x$ . The derivative of $-5$ is $0$ , and using the product rule for $yx^3$ , we get $3yx^2 + x^3\left(\frac{dy}{dx}\right)$ .
Equate Derivatives: Now we equate the derivatives from the left and right sides to get $(\frac{1}{2})(xy)^{-\frac{1}{2}} * (x\frac{dy}{dx} + y) = 3yx^2 + x^3\frac{dy}{dx}$ .
Solve for $\frac{dy}{dx}$ : We need to solve for $\frac{dy}{dx}$ . To do this, we'll collect all the terms containing $\frac{dy}{dx}$ on one side and the rest on the other side. This gives us $\frac{1}{2}(xy)^{-\frac{1}{2}} \cdot x\frac{dy}{dx} - x^3\frac{dy}{dx} = 3yx^2 - \frac{1}{2}(xy)^{-\frac{1}{2}} \cdot y$ .
Factor Out $(\frac{dy}{dx}):$ Factor out $(\frac{dy}{dx})$ from the terms on the left side to get (\frac{dy}{dx})((\frac{$1$}{$2$})(xy)^{-\frac{$1$}{$2$}} \cdot x - x^\(3) = $3$ yx^ $2$ - (\frac{ $1$ }{ $2$ })(xy)^{-\frac{ $1$ }{ $2$ }} \cdot y.
Isolate $(\frac{dy}{dx}):</b> Divide both sides by \$\left(\frac{1}{2}(xy)^{-\frac{1}{2}} \cdot x - x^3\right)$ to isolate $(\frac{dy}{dx})$ . This gives us (\frac{dy}{dx}) = \frac{$3$yx^$2$ - \left(\frac{$1$}{$2$}(xy)^{-\frac{$1$}{$2$}} \cdot y\right)}{\left(\frac{$1$}{$2$}(xy)^{-\frac{$1$}{$2$}} \cdot x - x^$3$\right)}.}
Simplify \((\frac{dy}{dx}): Simplify the expression for \$(\frac{dy}{dx}) if possible. However, in this case, the expression is already in its simplest form, so we have our final answer.

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Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-5+y x^{3}$

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Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-5+y x^{3}$