Find \; d y/d x of ≤ft(\; {x²}{a²} + \; {y²}{b²} = 1)

Correct Answer is : −b2xa2y{-b² x}{a² y}a2y−b2x

Correct Answer is : −b2xa2y{-b² x}{a² y}a2y−b2x

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Differential Calculus

Question : 30 of 69

Marks: +1, -0

Find

\; \frac{d y}{d x}

\left(\; \frac{x^{2}}{a^{2}} + \; \frac{y^{2}}{b^{2}} = 1\right)

Solution:

Concept:

Implicit differentiation of an ellipse equation.

Explanation:

Differentiate both sides with respect to

x

\frac{d}{dx}\left(\frac{x^2}{a^2}\right) + \frac{d}{dx}\left(\frac{y^2}{b^2}\right) = \frac{d}{dx}(1)

This gives

\frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0

Solve for

\frac{dy}{dx}

\frac{2y}{b^2} \cdot \frac{dy}{dx} = -\frac{2x}{a^2}

Thus

\frac{dy}{dx} = -\frac{2x}{a^2} \cdot \frac{b^2}{2y} = -\frac{b^2 x}{a^2 y}

Answer:

\frac{dy}{dx} = -\frac{b^2 x}{a^2 y}

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