Question

If y = x + e^x, find \[\frac{d^2 x}{d y^2}\] ?

Answer 1

Here,

\[y = x + e^x \]
\[ \Rightarrow \frac{d y}{d x} = 1 + e^x \]
\[ \Rightarrow \frac{dx}{dy} = \frac{1}{1 + e^x}\]
\[ \Rightarrow \frac{d^2 x}{d y^2} = \frac{- e^x}{\left( 1 + e^x \right)^2}\frac{dx}{dy} = \frac{- e^x}{\left( 1 + e^x \right)^3}\]