Practicing Success
If $y=x+e^x$, then $\frac{d^2 x}{d y^2}$ is equal to |
$\frac{1}{\left(1+e^x\right)^2}$ $-\frac{e^x}{\left(1+e^x\right)^2}$ $-\frac{e^x}{\left(1+e^x\right)^3}$ $e^x$ |
$-\frac{e^x}{\left(1+e^x\right)^3}$ |
We have, $y=x+e^x$ $\Rightarrow \frac{d y}{d x}=1+e^x$ $\Rightarrow \frac{d x}{d y}=\frac{1}{1+e^x}$ $\Rightarrow \frac{d^2 x}{d y^2}=\frac{d}{d y}\left(\frac{1}{1+e^x}\right)$ $\Rightarrow \frac{d^2 x}{d y^2}=-\frac{1}{\left(1+e^x\right)^2} \frac{d}{d y}\left(1+e^x\right)$ $\Rightarrow \frac{d^2 x}{d y^2}=-\frac{1}{\left(1+e^x\right)^2} e^x \frac{d x}{d y}=\frac{-e^x}{\left(1+e^x\right)^3}$ |