Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $-6 y^{3}-5 x^{4}+3 x+6 y-4 x y=5$

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Simplify expressions using trigonometric identities

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Q. Using implicit differentiation, find

\frac{d y}{d x}

\newline

-6 y^{3}-5 x^{4}+3 x+6 y-4 x y=5

Apply Chain Rule: We will differentiate each term of the equation with respect to $x$ , remembering to use the chain rule for terms involving $y$ , since $y$ is a function of $x$ .
Differentiate $-6y^3$ : Differentiate $-6y^3$ with respect to $x$ to get $-18y^2\left(\frac{dy}{dx}\right)$ .
Differentiate $-5x^4$ : Differentiate $-5x^4$ with respect to $x$ to get $-20x^3$ .
Differentiate $3x$ : Differentiate $3x$ with respect to $x$ to get $3$ .
Differentiate $6y$ : Differentiate $6y$ with respect to $x$ to get $6\frac{dy}{dx}$ .
Differentiate $-4xy$ : Differentiate $-4xy$ with respect to $x$ . This requires the product rule: the derivative of the first function ( $-4y$ ) times the second function ( $x$ ) plus the first function times the derivative of the second function ( $1$ ). This gives us $-4y - 4x\frac{dy}{dx}$ .
Differentiate constant $5$ : Differentiate the constant $5$ with respect to $x$ to get $0$ .
Combine differentiated terms: Combine all the differentiated terms to form the new equation: $-18y^2\frac{dy}{dx} - 20x^3 + 3 + 6\frac{dy}{dx} - 4y - 4x\frac{dy}{dx} = 0$ .
Collect terms: Now, we collect all the terms with $\frac{dy}{dx}$ on one side and the rest on the other side: $-18y^2\frac{dy}{dx} + 6\frac{dy}{dx} - 4x\frac{dy}{dx} = 20x^3 - 3 + 4y$ .
Factor out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ from the left side of the equation: $\frac{dy}{dx}(-18y^2 + 6 - 4x) = 20x^3 - 3 + 4y$ .
Solve for $(\frac{dy}{dx}):</b> Solve for \$(\frac{dy}{dx})$ by dividing both sides of the equation by $(-18y^2 + 6 - 4x)$ : $(\frac{dy}{dx}) = \frac{20x^3 - 3 + 4y}{-18y^2 + 6 - 4x}$ .

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Using implicit differentiation, find dydx \frac{d y}{d x} dxdy​.\newline−6y3−5x4+3x+6y−4xy=5 -6 y^{3}-5 x^{4}+3 x+6 y-4 x y=5 −6y3−5x4+3x+6y−4xy=5

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $-6 y^{3}-5 x^{4}+3 x+6 y-4 x y=5$