If the solve the problem

Question:

If $y=x+e^{x}$, find $\frac{d^{2} x}{d y^{2}}$

Solution:

Given:

$y=x+e^{x}$

$\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{d}^{2} \mathrm{y}}=\frac{1}{\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}}$

$\frac{d y}{d x}=1+e^{x}$

$\frac{d^{2} y}{d x^{2}}=e^{x}$

$\frac{d^{2} x}{d^{2} y}=\frac{1}{e^{x}}$

$=e^{-x}$

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