To determine [tex]\(\frac{dy}{dx}\)[/tex] at the point [tex]\((1,1)\)[/tex] for the given equation [tex]\(x^3 y^3 + x = y\)[/tex], follow these detailed steps:
1. Differentiate Implicitly:
Start by differentiating both sides of the equation with respect to [tex]\(x\)[/tex].
[tex]\[ x^3 y^3 + x = y \][/tex]
When differentiating implicitly, remember that [tex]\(y\)[/tex] is a function of [tex]\(x\)[/tex] (i.e., [tex]\(y\)[/tex] depends on [tex]\(x\)[/tex]).
Differentiating [tex]\(x^3 y^3\)[/tex] with respect to [tex]\(x\)[/tex] using the product rule:
[tex]\[ \frac{d}{dx}(x^3 y^3) = \frac{d}{dx}(x^3) \cdot y^3 + x^3 \cdot \frac{d}{dx}(y^3) \][/tex]
Since [tex]\(\frac{d}{dx}(y^3) = 3y^2 \cdot \frac{dy}{dx}\)[/tex], we have
[tex]\[ \frac{d}{dx}(x^3 y^3) = 3x^2 y^3 + x^3 \cdot 3y^2 \frac{dy}{dx} = 3x^2 y^3 + 3x^3 y^2 \frac{dy}{dx} \][/tex]
Differentiating [tex]\(x\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{d}{dx}(x) = 1 \][/tex]
Differentiating [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[ \frac{d}{dx}(y) = \frac{dy}{dx} \][/tex]
Combining these terms, the differentiated equation is:
[tex]\[ 3x^2 y^3 + 3x^3 y^2 \frac{dy}{dx} + 1 = \frac{dy}{dx} \][/tex]
2. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
Rearrange the equation to gather terms involving [tex]\(\frac{dy}{dx}\)[/tex] on one side:
[tex]\[ 3x^3 y^2 \frac{dy}{dx} - \frac{dy}{dx} = -3x^2 y^3 - 1 \][/tex]
Factor out [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ (3x^3 y^2 - 1) \frac{dy}{dx} = -3x^2 y^3 - 1 \][/tex]
Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{-3x^2 y^3 - 1}{3x^3 y^2 - 1} \][/tex]
3. Evaluate at [tex]\((x, y) = (1, 1)\)[/tex]:
Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 1\)[/tex] into the derivative expression:
[tex]\[ \frac{dy}{dx} \bigg|_{(1,1)} = \frac{-3(1)^2(1)^3 - 1}{3(1)^3(1)^2 - 1} \][/tex]
Simplify the terms:
[tex]\[ \frac{dy}{dx} \bigg|_{(1,1)} = \frac{-3(1) - 1}{3(1) - 1} = \frac{-3 - 1}{3 - 1} = \frac{-4}{2} = -2 \][/tex]
Thus, the value of [tex]\(\frac{dy}{dx}\)[/tex] at [tex]\((1,1)\)[/tex] is [tex]\(-2\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-2} \][/tex]
