If `e^y(x+1)=1,` show that `(d^(2y))/(dx^2)=((dy)/(dx))^2` If `y=sin(2sin^(-1)x),` show that `((1-x^2)d^(2y))/(dx^2)=x(dy)/(dx)-4y`

Explanation

Let's solve the problem step by step.

### Part 1: Show that \( \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2 \) given \( e^y(x+1) = 1 \).

1. **Start with the given equation**:
\[ e^y(x + 1) = 1 \]
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