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

sqrt(xy)=-2+yx^(3)

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-2+y x^{3}$

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Find derivatives of radical functions

Full solution

Q. Using implicit differentiation, find

\frac{d y}{d x}

.

\newline

\sqrt{x y}=-2+y x^{3}

Identify Equation: Identify the equation that needs to be differentiated implicitly. $\newline$ Equation: $\sqrt{xy} = -2 + yx^{3}$
Differentiate Both Sides: Differentiate both sides of the equation with respect to $x$ , remembering to apply the product rule to $xy$ and $yx^{3}$ , and the chain rule to $\sqrt{xy}$ . $\frac{d}{dx}(\sqrt{xy}) = \frac{d}{dx}(-2 + yx^{3})$
Apply Chain Rule: Apply the chain rule to the left side: $\frac{d}{dx}(\sqrt{u}) = \frac{1}{2}u^{-\frac{1}{2}} \cdot \frac{du}{dx}$ , where $u = xy$ . $\frac{d}{dx}(\sqrt{xy}) = \frac{1}{2}(xy)^{-\frac{1}{2}} \cdot \frac{d}{dx}(xy)$
Apply Product Rule: Apply the product rule to $\frac{d}{dx}(xy)$ : $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x$ and $v = y$ .
$\frac{d}{dx}(xy) = \frac{d}{dx}(x)y + x*\frac{d}{dx}(y)$
$\frac{d}{dx}(xy) = y + x\left(\frac{dy}{dx}\right)$
Substitute Result: Substitute the result from the product rule into the chain rule result. $\newline$ $(\frac{1}{2})(xy)^{-\frac{1}{2}} \cdot (y + x\frac{dy}{dx})$
Differentiate Right Side: Differentiate the right side of the equation with respect to $x$ , applying the product rule to $yx^{3}$ .
$\frac{d}{dx}(-2 + yx^{3}) = \frac{d}{dx}(-2) + \frac{d}{dx}(yx^{3})$
$\frac{d}{dx}(-2 + yx^{3}) = 0 + \frac{d}{dx}(yx^{3})$
Apply Product Rule: Apply the product rule to $\frac{d}{dx}(yx^{3})$ : $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = y$ and $v = x^{3}$ .
$\frac{d}{dx}(yx^{3}) = \frac{d}{dx}(y)x^{3} + y*\frac{d}{dx}(x^{3})$
$\frac{d}{dx}(yx^{3}) = \left(\frac{dy}{dx}\right)x^{3} + y*3x^{2}$
Combine Differentiated Results: Combine the differentiated results and set them equal to each other. $\newline$ $(\frac{1}{2})(xy)^{-\frac{1}{2}} * (y + x\frac{dy}{dx}) = 0 + \frac{dy}{dx}x^{3} + y\cdot 3x^{2}$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ by isolating terms involving $\frac{dy}{dx}$ on one side of the equation. $\newline$ $\left(\frac{1}{2}\right)(xy)^{-\frac{1}{2}} \cdot y + \left(\frac{1}{2}\right)(xy)^{-\frac{1}{2}} \cdot x\left(\frac{dy}{dx}\right) = \left(\frac{dy}{dx}\right)x^{3} + y\cdot 3x^{2}$
Subtract Terms: Subtract $\frac{dy}{dx}x^{3}$ from both sides to get all $\frac{dy}{dx}$ terms on one side. $\newline$ $\left(\frac{1}{2}\right)(xy)^{-\frac{1}{2}} \cdot x\frac{dy}{dx} - \frac{dy}{dx}x^{3} = y\cdot 3x^{2} - \left(\frac{1}{2}\right)(xy)^{-\frac{1}{2}} \cdot y$
Factor out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ on the left side of the equation. $\frac{dy}{dx} \times \left(\frac{1}{2}(xy)^{-\frac{1}{2}} \times x - x^{3}\right) = y\times3x^{2} - \frac{1}{2}(xy)^{-\frac{1}{2}} \times y$
Divide Both Sides: Divide both sides by $[(1/2)(xy)^{-1/2} \cdot x - x^{3}]$ to solve for $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{y\cdot 3x^{2} - (1/2)(xy)^{-1/2} \cdot y}{(1/2)(xy)^{-1/2} \cdot x - x^{3}}$

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