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Using implicit differentiation, find 
(dy)/(dx).

sqrt(xy)=7+x^(2)y^(2)

Using implicit differentiation, find dydx \frac{d y}{d x} .\newlinexy=7+x2y2 \sqrt{x y}=7+x^{2} y^{2}

Full solution

Q. Using implicit differentiation, find dydx \frac{d y}{d x} .\newlinexy=7+x2y2 \sqrt{x y}=7+x^{2} y^{2}
  1. Differentiate with respect to xx: Differentiate both sides of the equation with respect to xx using implicit differentiation.\newlineThe left side of the equation is xy\sqrt{xy}, which is (xy)1/2(xy)^{1/2}. Using the chain rule, the derivative of (xy)1/2(xy)^{1/2} with respect to xx is (1/2)(xy)1/2(y+xdydx)(1/2)(xy)^{-1/2} * (y + x\frac{dy}{dx}).\newlineThe right side of the equation is 7+x2y27 + x^2 * y^2. The derivative of 77 with respect to xx is xx00. The derivative of xx11 with respect to xx is xx33 using the product rule.\newlineSo, differentiating both sides gives us:\newlinexx44.
  2. Simplify the equation: Simplify the differentiated equation.\newline(12)(xy)12(y+xdydx)=2xy2+2x2ydydx(\frac{1}{2})(xy)^{-\frac{1}{2}} * (y + x\frac{dy}{dx}) = 2x * y^2 + 2x^2 * y\frac{dy}{dx}.\newlineMultiply both sides by 2(xy)122(xy)^{\frac{1}{2}} to get rid of the fraction and the square root:\newliney+xdydx=4xy2(xy)12+4x2y(xy)12dydxy + x\frac{dy}{dx} = 4x * y^2 * (xy)^{\frac{1}{2}} + 4x^2 * y * (xy)^{\frac{1}{2}} * \frac{dy}{dx}.
  3. Isolate terms: Isolate terms with dydx\frac{dy}{dx} on one side and the rest on the other side.xdydx4x2y(xy)12dydx=4xy2(xy)12yx\frac{dy}{dx} - 4x^2 y (xy)^{\frac{1}{2}} \frac{dy}{dx} = 4x y^2 (xy)^{\frac{1}{2}} - y.
  4. Factor out dydx\frac{dy}{dx}: Factor out dydx\frac{dy}{dx} on the left side of the equation.\newlinedydx(x4x2y(xy)12)=4xy2(xy)12y\frac{dy}{dx}(x - 4x^2 \cdot y \cdot (xy)^{\frac{1}{2}}) = 4x \cdot y^2 \cdot (xy)^{\frac{1}{2}} - y.
  5. Solve for (dydx):(\frac{dy}{dx}): Solve for (dydx).(\frac{dy}{dx}).dydx=4xy2(xy)12yx4x2y(xy)12.\frac{dy}{dx} = \frac{4x \cdot y^2 \cdot (xy)^{\frac{1}{2}} - y}{x - 4x^2 \cdot y \cdot (xy)^{\frac{1}{2}}}.

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