If $ y=sinx+e^x$, then $\frac{d^2x}{dy^2}$ is equal to :

$\frac{sin\, x-e^x}{(cos\, x +e^x)^3}$

The correct answer is Option (2) → $\frac{sin\, x-e^x}{(cos\, x +e^x)^3}$

$y=\sin x+e^x$

so $\frac{dy}{dx}=\cos x+e^x⇒\frac{dx}{dy}=\frac{1}{\cos x+e^x}$

$⇒\frac{d^2x}{dy^2}=\frac{-1(-\sin x+e^x)}{(\cos x+e^x)^2}\frac{dx}{dy}$

$=\frac{\sin x-e^x}{(\cos x +e^x)^3}$