Question Statement

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

$y^4-y^2=10x-3$

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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 $x$. Since $y$ depends on $x$, we apply the chain rule when differentiating any terms involving $y$, multiplying by $\frac{dy}{dx}$.

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Solution

We begin with the given equation and differentiate both sides with respect to $x$.

Step 1: Differentiate both sides

Differentiate each term with respect to $x$:

$\frac{d}{dx}\left(y^4\right)-\frac{d}{dx}\left(y^2\right)=10\frac{d}{dx}(x)-\frac{d}{dx}(3)$

Step 2: Apply the chain rule

For terms involving $y$, we apply the chain rule (power rule + chain rule):

• $\frac{d}{dx}(y^4)=4y^3\cdot \frac{dy}{dx}$
• $\frac{d}{dx}(y^2)=2y\cdot \frac{dy}{dx}$

For the right side:

• $\frac{d}{dx}(10x)=10$
• $\frac{d}{dx}(3)=0$

This yields:

$4y^3\frac{dy}{dx}-2y\frac{dy}{dx}=10$

Step 3: Factor out $\frac{dy}{dx}$

Collect all terms containing $\frac{dy}{dx}$ on the left side:

$\left(4y^3-2y\right)\frac{dy}{dx}=10$

Step 4: Solve for $\frac{dy}{dx}$

Divide both sides by the coefficient $(4y^3-2y)$:

$\frac{dy}{dx}=\frac{10}{4y^3-2y}$

Step 5: Simplify the expression

Factor out 2 from the denominator:

$\frac{dy}{dx}=\frac{10}{2\left(2y^3-y\right)}=\frac{5}{2y^3-y}$

Further factor the denominator by extracting $y$:

$\frac{dy}{dx}=\frac{5}{y\left(2y^2-1\right)}$

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Key Formulas or Methods Used

• Implicit Differentiation: Differentiating both sides of an equation with respect to $x$, treating $y$ as a function of $x$
• Chain Rule: $\frac{d}{dx}[f(y)]=f^{\prime }(y)\cdot \frac{dy}{dx}$
• Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1}$ (extended to $y$ terms via chain rule)
• Algebraic Simplification: Factoring to simplify complex rational expressions

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Summary of Steps

1. Differentiate both sides of the equation with respect to $x$.
2. Apply chain rule to $y$ terms (multiply by $\frac{dy}{dx}$).
3. Simplify the right-hand side constants.
4. Factor $\frac{dy}{dx}$ from all terms on the left side.
5. Isolate $\frac{dy}{dx}$ by dividing both sides by its coefficient.
6. Simplify the resulting fraction by factoring common terms in the numerator and denominator.