QuestionJuly 14, 2025

Find (dy)/(dx) at the given point. 7x^2-3y^3=4x-3y at (0,1) The value of (dy)/(dx) at (0,1) is (1)/(3) (Simplify your answer.)

Find (dy)/(dx) at the given point. 7x^2-3y^3=4x-3y at (0,1) The value of (dy)/(dx) at (0,1) is (1)/(3) (Simplify your answer.)
Find (dy)/(dx) at the given point. 7x^2-3y^3=4x-3y at (0,1) The value of (dy)/(dx) at (0,1) is (1)/(3) (Simplify your answer.)

Solution
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Answer

\frac{1}{3} Explanation 1. Differentiate both sides with respect to x Differentiate implicitly: 14x - 9y^2 \frac{dy}{dx} = 4 - 3 \frac{dy}{dx}. 2. Solve for \frac{dy}{dx} Rearrange terms: 9y^2 \frac{dy}{dx} + 3 \frac{dy}{dx} = 4 - 14x. Factor out \frac{dy}{dx}: \frac{dy}{dx}(9y^2 + 3) = 4 - 14x. Solve: \frac{dy}{dx} = \frac{4 - 14x}{9y^2 + 3}. 3. Substitute the point (0,1) Substitute x=0 and y=1: \frac{dy}{dx} = \frac{4 - 14(0)}{9(1)^2 + 3} = \frac{4}{12} = \frac{1}{3}.

Explanation

1. Differentiate both sides with respect to x<br /> Differentiate implicitly: $14x - 9y^2 \frac{dy}{dx} = 4 - 3 \frac{dy}{dx}$.<br />2. Solve for $\frac{dy}{dx}$<br /> Rearrange terms: $9y^2 \frac{dy}{dx} + 3 \frac{dy}{dx} = 4 - 14x$. Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(9y^2 + 3) = 4 - 14x$. Solve: $\frac{dy}{dx} = \frac{4 - 14x}{9y^2 + 3}$.<br />3. Substitute the point (0,1)<br /> Substitute $x=0$ and $y=1$: $\frac{dy}{dx} = \frac{4 - 14(0)}{9(1)^2 + 3} = \frac{4}{12} = \frac{1}{3}$.
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