Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $-y-7 y^{3}-4 x^{4}+6 x-2 x y=1$

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Simplify expressions using trigonometric identities

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Q. Using implicit differentiation, find

\frac{d y}{d x}

\newline

-y-7 y^{3}-4 x^{4}+6 x-2 x y=1

Apply Differentiation Rules: We will differentiate each term of the equation with respect to $x$ , remembering to use the product rule for the term $-2xy$ and the chain rule for terms involving $y$ , since $y$ is a function of $x$ .
Differentiate Terms: Differentiate $-y$ with respect to $x$ to get $-\frac{dy}{dx}$ .
Combine Differentiated Terms: Differentiate $-7y^3$ with respect to $x$ to get $-21y^2\frac{dy}{dx}$ , using the chain rule.
Group Terms: Differentiate $-4x^4$ with respect to $x$ to get $-16x^3$ .
Factor out $\frac{dy}{dx}$ : Differentiate $6x$ with respect to $x$ to get $6$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .Differentiate the constant term $1$ with respect to $x$ to get $0$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .Differentiate the constant term $1$ with respect to $x$ to get $0$ .Combine all the differentiated terms to form the equation: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 16x^3 + 6 - 2y - 2x\frac{dy}{dx} = 0$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .Differentiate the constant term $1$ with respect to $x$ to get $0$ .Combine all the differentiated terms to form the equation: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 16x^3 + 6 - 2y - 2x\frac{dy}{dx} = 0$ .Group all the terms involving $\frac{dy}{dx}$ on one side and the rest on the other side to get: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 2x\frac{dy}{dx} = 16x^3 - 6 + 2y$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .Differentiate the constant term $1$ with respect to $x$ to get $0$ .Combine all the differentiated terms to form the equation: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 16x^3 + 6 - 2y - 2x\frac{dy}{dx} = 0$ .Group all the terms involving $\frac{dy}{dx}$ on one side and the rest on the other side to get: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 2x\frac{dy}{dx} = 16x^3 - 6 + 2y$ .Factor out $\frac{dy}{dx}$ from the left side to get: $-2xy$ $1$ .
Solve for $\frac{dy}{dx}$ : Differentiate $-2xy$ with respect to $x$ using the product rule to get $-2y - 2x\frac{dy}{dx}$ .Differentiate the constant term $1$ with respect to $x$ to get $0$ .Combine all the differentiated terms to form the equation: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 16x^3 + 6 - 2y - 2x\frac{dy}{dx} = 0$ .Group all the terms involving $\frac{dy}{dx}$ on one side and the rest on the other side to get: $-\frac{dy}{dx} - 21y^2\frac{dy}{dx} - 2x\frac{dy}{dx} = 16x^3 - 6 + 2y$ .Factor out $\frac{dy}{dx}$ from the left side to get: $-2xy$ $1$ .Solve for $\frac{dy}{dx}$ by dividing both sides by $-2xy$ $3$ to get: $-2xy$ $4$ .