Question

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

• A. $\frac{1}{{(1+e^x)}^2}$
• B. $\frac{-e^x}{{(1+e^x)}^3}$
• C. $\frac{-e^x}{{(1+e^x)}^2}$
• D. $e^x$

Answer 1

The correct option is B $\frac{-e^x}{{(1+e^x)}^3}$

$y=x+e^x$

Differentiating both sides w.r.t $x,$ we get:

$\Rightarrow$ $\frac{dy}{dx}=1+e^x$

Taking reciprocal on both sides,

$\Rightarrow$ $\frac{dx}{dy}=\frac{1}{1+e^x}$ ---- ( 1 )

Differentiating both sides w.r.t. $y,$ we get:

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

From ( 1 ), we get:

$\Rightarrow$ $\frac{d^{2}x}{d{y}^2}=\frac{-e^x}{(1+e^x{)}^3}$