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Using implicit differentiation, find $\frac{dy}{dx}$ . $\sin(xy) = 2x^{4}y^{3} + 4$

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Find derivatives of radical functions

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Q. Using implicit differentiation, find

\frac{dy}{dx}

.

\sin(xy) = 2x^{4}y^{3} + 4

Apply Chain and Product Rule: Differentiate both sides of the equation with respect to $x$ using implicit differentiation. $\newline$ The left side of the equation is $\sin(xy)$ , which is a product of $x$ and $y$ , so we will need to use the product rule in combination with the chain rule. The right side of the equation is $2x^4y^3 + 4$ , which is a sum of terms involving products of $x$ and $y$ , so we will also need to use the product rule there.
Differentiate $\sin(xy)$ : Differentiate the left side, $\sin(xy)$ , with respect to $x$ . Using the chain rule, the derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$ , and then we multiply by the derivative of $u$ with respect to $x$ , where $u = xy$ . This gives us $\cos(xy) \cdot (y + x\frac{dy}{dx})$ .
Differentiate $2x^4y^3 + 4$ : Differentiate the right side, $2x^4y^3 + 4$ , with respect to $x$ . Using the product rule, the derivative of $2x^4y^3$ with respect to $x$ is $2 \times (4x^3y^3 + x^4 \times 3y^2\frac{dy}{dx})$ . The derivative of the constant $4$ with respect to $x$ is $0$ .
Write Differentiated Equation: Write down the differentiated equation. $\newline$ $\cos(xy) \cdot (y + x\frac{dy}{dx}) = 2 \cdot (4x^3y^3 + x^4 \cdot 3y^2\frac{dy}{dx}) + 0$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ . We need to collect all terms involving $\frac{dy}{dx}$ on one side and the rest on the other side to solve for $\frac{dy}{dx}$ . $\cos(xy) \cdot x\frac{dy}{dx} - 2x^4 \cdot 3y^2\frac{dy}{dx} = 2 \cdot 4x^3y^3 - \cos(xy) \cdot y$
Factor out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ from the terms on the left side. $\newline$ $\frac{dy}{dx}(x \cdot \cos(xy) - 2x^4 \cdot 3y^2) = 2 \cdot 4x^3y^3 - \cos(xy) \cdot y$
Isolate $\frac{dy}{dx}$ : Isolate $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{2 \cdot 4x^3y^3 - \cos(xy) \cdot y}{x \cdot \cos(xy) - 2x^4 \cdot 3y^2}$
Simplify (\frac{dy}{dx}): Simplify the expression for \$(\frac{dy}{dx}).\(\newline\$(\frac{dy}{dx}) = \frac{8x^3y^3 - \cos(xy) \cdot y}{x \cdot \cos(xy) - 6x^4y^2}\)

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