If y=x+e^x, find d^2x dy^2 . - Vidyadip

Question

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

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Answer

Here,
$\text{y}=\text{x}+\text{e}^\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=1+\text{e}^\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{1+\text{e}^\text{x}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\text{e}^\text{x}}{(1+\text{e}^\text{x})^2}$
$\frac{\text{dx}}{\text{dy}}=-\frac{-\text{e}^\text{x}}{(1+\text{e}^\text{x})^3}$

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