Question Statement

Find $\frac{dy}{dx}$ for the curve defined implicitly by the equation:

$4x^2+y^2=8$

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Background and Explanation

This problem requires implicit differentiation, a technique used when $y$ is defined as a function of $x$ implicitly rather than explicitly solved for $y$. Since $y$ depends on $x$, we apply the chain rule when differentiating terms containing $y$, then solve algebraically for $\frac{dy}{dx}$.

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Solution

We begin with the given equation of an ellipse:

$4x^2+y^2=8$

Differentiate both sides of the equation with respect to $x$:

$\begin{array}{rl}4\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(y^2\right)& =\frac{d}{dx}(8)\\ 4(2x)+2y^{2-1}\frac{dy}{dx}& =0\\ 8x+2y\frac{dy}{dx}& =0\end{array}$

Next, isolate the term containing $\frac{dy}{dx}$ by moving $8x$ to the right side:

$2y\frac{dy}{dx}=-8x$

Solve for $\frac{dy}{dx}$ by dividing both sides by $2y$:

$\frac{dy}{dx}=\frac{-8x}{2y}$

Simplify the fraction by dividing numerator and denominator by $2$:

$\frac{dy}{dx}=\frac{-4x}{y}$

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Key Formulas or Methods Used

• Implicit Differentiation: Differentiating both sides of an equation with respect to $x$ while treating $y$ as a function of $x$
• Chain Rule: $\frac{d}{dx}[f(y)]=f^{\prime }(y)\cdot \frac{dy}{dx}$, specifically $\frac{d}{dx}(y^2)=2y\cdot \frac{dy}{dx}$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for differentiating $x^2$
• Constant Rule: $\frac{d}{dx}(c)=0$ for any constant $c$

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Summary of Steps

1. Start with the given equation: $4x^2+y^2=8$.
2. Differentiate both sides with respect to $x$.
3. Apply the power rule to $4x^2$ to obtain $8x$.
4. Apply the chain rule to $y^2$ to obtain $2y\frac{dy}{dx}$.
5. Set the derivative of the constant $8$ equal to $0$.
6. Rearrange the equation to isolate the term with $\frac{dy}{dx}$: $2y\frac{dy}{dx}=-8x$.
7. Divide both sides by $2y$ to solve for $\frac{dy}{dx}$.
8. Simplify the resulting expression to get the final answer: $\frac{dy}{dx}=\frac{-4x}{y}$.