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

View step-by-step help

Home

Math Problems

Algebra 2

Simplify expressions using trigonometric identities

Full solution

Q. Using implicit differentiation, find

\frac{d y}{d x}

\newline

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

Apply Implicit Differentiation: To find the derivative of $y$ with respect to $x$ , we will use implicit differentiation on the given equation $-2xy^{4} - 6xy^{2} = 3x + 4$ .
Use Product and Chain Rule: Differentiate both sides of the equation with respect to $x$ . Remember that $y$ is a function of $x$ , so we need to apply the product rule to the terms involving $y$ and the chain rule to the derivatives of $y$ .
Combine and Simplify Equations: Differentiating the left side of the equation, we get the derivative of $-2xy^{4}$ with respect to $x$ as $-2[y^{4} + 4xy^{3}(\frac{dy}{dx})]$ using the product rule and chain rule. Similarly, the derivative of $-6xy^{2}$ with respect to $x$ is $-6[y^{2} + 2xy(\frac{dy}{dx})]$ .
Group Terms and Solve: Differentiating the right side of the equation with respect to $x$ gives us $3$ , since the derivative of $3x$ with respect to $x$ is $3$ and the derivative of a constant ( $4$ ) is $0$ .
Factor Out $\frac{dy}{dx}$ : Now we combine the derivatives from the previous steps to form the equation: $-2[y^{4} + 4xy^{3}\left(\frac{dy}{dx}\right)] - 6[y^{2} + 2xy\left(\frac{dy}{dx}\right)] = 3$ .
Solve for $\frac{dy}{dx}$ : We simplify the equation by distributing the $-2$ and $-6$ to get: $-2y^{4} - 8xy^{3}\frac{dy}{dx} - 6y^{2} - 12xy\frac{dy}{dx} = 3$ .
Further Simplify Expression: Next, we group the terms with $\frac{dy}{dx}$ on one side and the rest on the other side to solve for $\frac{dy}{dx}$ . This gives us $-8xy^{3}\frac{dy}{dx} - 12xy\frac{dy}{dx} = 3 + 2y^{4} + 6y^{2}$ .
Further Simplify Expression: Next, we group the terms with $\frac{dy}{dx}$ on one side and the rest on the other side to solve for $\frac{dy}{dx}$ . This gives us $-8xy^{3}\frac{dy}{dx} - 12xy\frac{dy}{dx} = 3 + 2y^{4} + 6y^{2}$ .We factor out $\frac{dy}{dx}$ from the terms on the left side to get $\frac{dy}{dx}(-8xy^{3} - 12xy) = 3 + 2y^{4} + 6y^{2}$ .
Further Simplify Expression: Next, we group the terms with $\frac{dy}{dx}$ on one side and the rest on the other side to solve for $\frac{dy}{dx}$ . This gives us $-8xy^{3}\frac{dy}{dx} - 12xy\frac{dy}{dx} = 3 + 2y^{4} + 6y^{2}$ .We factor out $\frac{dy}{dx}$ from the terms on the left side to get $\frac{dy}{dx}(-8xy^{3} - 12xy) = 3 + 2y^{4} + 6y^{2}$ .Now we solve for $\frac{dy}{dx}$ by dividing both sides by $(-8xy^{3} - 12xy)$ to get $\frac{dy}{dx} = \frac{3 + 2y^{4} + 6y^{2}}{-8xy^{3} - 12xy}$ .
Further Simplify Expression: Next, we group the terms with $\frac{dy}{dx}$ on one side and the rest on the other side to solve for $\frac{dy}{dx}$ . This gives us $-8xy^{3}\frac{dy}{dx} - 12xy\frac{dy}{dx} = 3 + 2y^{4} + 6y^{2}$ . We factor out $\frac{dy}{dx}$ from the terms on the left side to get $\frac{dy}{dx}(-8xy^{3} - 12xy) = 3 + 2y^{4} + 6y^{2}$ . Now we solve for $\frac{dy}{dx}$ by dividing both sides by $(-8xy^{3} - 12xy)$ to get $\frac{dy}{dx} = \frac{3 + 2y^{4} + 6y^{2}}{-8xy^{3} - 12xy}$ . We can simplify the expression further by factoring out a $2$ from the numerator and a $-4x$ from the denominator to get $\frac{dy}{dx}$ $0$ .