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

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

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=7+x^{2} y^{2}$

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Find trigonometric ratios of special angles

Full solution

Q. Using implicit differentiation, find

\frac{d y}{d x}

.

\newline

\sqrt{x y}=7+x^{2} y^{2}

Differentiate with respect to $x$ : Differentiate both sides of the equation with respect to $x$ using implicit differentiation. $\newline$ The left side of the equation is $\sqrt{xy}$ , which is $(xy)^{1/2}$ . Using the chain rule, the derivative of $(xy)^{1/2}$ with respect to $x$ is $(1/2)(xy)^{-1/2} * (y + x\frac{dy}{dx})$ . $\newline$ The right side of the equation is $7 + x^2 * y^2$ . The derivative of $7$ with respect to $x$ is $x$ $0$ . The derivative of $x$ $1$ with respect to $x$ is $x$ $3$ using the product rule. $\newline$ So, differentiating both sides gives us: $\newline$ $x$ $4$ .
Simplify the equation: Simplify the differentiated equation. $\newline$ $(\frac{1}{2})(xy)^{-\frac{1}{2}} * (y + x\frac{dy}{dx}) = 2x * y^2 + 2x^2 * y\frac{dy}{dx}$ . $\newline$ Multiply both sides by $2(xy)^{\frac{1}{2}}$ to get rid of the fraction and the square root: $\newline$ $y + x\frac{dy}{dx} = 4x * y^2 * (xy)^{\frac{1}{2}} + 4x^2 * y * (xy)^{\frac{1}{2}} * \frac{dy}{dx}$ .
Isolate terms: Isolate terms with $\frac{dy}{dx}$ on one side and the rest on the other side. $x\frac{dy}{dx} - 4x^2 y (xy)^{\frac{1}{2}} \frac{dy}{dx} = 4x y^2 (xy)^{\frac{1}{2}} - y$ .
Factor out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ on the left side of the equation. $\newline$ $\frac{dy}{dx}(x - 4x^2 \cdot y \cdot (xy)^{\frac{1}{2}}) = 4x \cdot y^2 \cdot (xy)^{\frac{1}{2}} - y$ .
Solve for $(\frac{dy}{dx}):$ Solve for $(\frac{dy}{dx}).$ $\frac{dy}{dx} = \frac{4x \cdot y^2 \cdot (xy)^{\frac{1}{2}} - y}{x - 4x^2 \cdot y \cdot (xy)^{\frac{1}{2}}}.$

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