Question:
Let \(y = e^{2x}\). Then \(\frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2}\) is:
Let \(y = e^{2x}\). Then \(\frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2}\) is:
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Use inverse differentiation identities carefully.
Updated On: Mar 23, 2026
- \(1\)
- \(e^{-2x}\)
- \(2e^{-2x}\)
- -2e⁻2x
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The Correct Option is A
Solution and Explanation
Step 1:
\[ \frac{dy}{dx} = 2 e^{2x}, \quad \frac{d^2 y}{dx^2} = 4 e^{2x} \]
Step 2:
\[ \frac{dx}{dy} = \frac{1}{2} e^{-2x}, \quad \frac{d^2 x}{dy^2} = -\frac{1}{4} e^{-4x} \]
Step 3:
\[ \frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2} = 4 e^{2x} \cdot \left(-\frac{1}{4} e^{-4x}\right) = - e^{-2x} \]
\[ \frac{dy}{dx} = 2 e^{2x}, \quad \frac{d^2 y}{dx^2} = 4 e^{2x} \]
Step 2:
\[ \frac{dx}{dy} = \frac{1}{2} e^{-2x}, \quad \frac{d^2 x}{dy^2} = -\frac{1}{4} e^{-4x} \]
Step 3:
\[ \frac{d^2 y}{dx^2} \cdot \frac{d^2 x}{dy^2} = 4 e^{2x} \cdot \left(-\frac{1}{4} e^{-4x}\right) = - e^{-2x} \]
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