Question

If \( x^{\frac{2}{5}} + y^{\frac{2}{5}} = \text{a}^{\frac{2}{5}} \) then \( \frac{dy}{dx} = \)

• A. \( \sqrt[5]{\left(\frac{y}{x}\right)^3} \)
• B. \( -\sqrt[5]{\left(\frac{x}{y}\right)^3} \)
• C. \( \sqrt[5]{\left(\frac{x}{y}\right)^3} \)
• D. \( -\sqrt[5]{\left(\frac{y}{x}\right)^3} \)

Hint:

For equations of type \( x^n + y^n = a^n \), the derivative is \( -\frac{x^{n-1}}{y^{n-1}} \).

Answer 1

The Correct Option is D

Step 1: Concept Use implicit differentiation: differentiate both sides with respect to \( x \).

Step 2: Meaning \(\frac{d}{dx}(x^{2/5}) + \frac{d}{dx}(y^{2/5}) = \frac{d}{dx}(a^{2/5})\).

Step 3: Analysis \( \frac{2}{5}x^{-3/5} + \frac{2}{5}y^{-3/5} \frac{dy}{dx} = 0 \). \( y^{-3/5} \frac{dy}{dx} = -x^{-3/5} \). \( \frac{dy}{dx} = -\frac{x^{-3/5}}{y^{-3/5}} = -\frac{y^{3/5}}{x^{3/5}} \).

Step 4: Conclusion Rearranging into root form gives \( \frac{dy}{dx} = -\sqrt[5]{\left(\frac{y}{x}\right)^3} \). Final Answer: (D)