Question

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

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

Answer 1

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

$y=x+e^x$
Differentiate w.r.t. 'y'
$1=\frac{dx}{dy}+e^x\frac{dx}{dy}$
$1=(1+e^x)\frac{dx}{dy}$
$\frac{dx}{dy}=\frac{1}{(1+e^x)}$.................(1)
$\frac{d^{2}x}{d{y}^2}=\frac{-e^x\frac{dx}{dy}}{{(1+e^x)}^2}$
$\frac{d^{2}x}{d{y}^2}=\frac{-e^x}{{(1+e^x)}^2}\times \frac{1}{(1+e^x)}$ ..............from (1)
$\frac{d^{2}x}{d{y}^2}=\frac{-e^x}{{(1+e^x)}^3}$