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Question:

If ey(x+1)=1, then show that d²y/dx²=(dy/dx)²

Solution:

ey(x+1)=1
ey × 1 + [ey × dy/dx × (x+1)] = 0
Differentiating on both sides ⇒ dy/dx = -ey/ey × (x+1) = -1/(x+1)
Differentiating again ⇒ d²y/dx² = -[-1/(x+1)²] = 1/(x+1)²
⇒ d²y/dx² = 1/(x+1)² = (-1/(x+1))² = (dy/dx)²
Hence proved