Question

If x = e^2y, then `("d"^2y)/"dx"^2*("d"^2x)/"dy"^2` is equal to

Options

• e^-2y

• -2e^-2y

• 2e^-2y

• 1

Answer 1

-2e^-2y 

Explanation:

x = e^2y

∴ `"dx"/"dy"` = 2e^2y

∴ `("d"^2x)/"dy"^2` = 4e^2y

Now, x = e^2y

∴ log x = 2y

∴ y = `1/2` log x

∴ `"dy"/"dx" = 1/"2x"`

∴ `("d"^2x)/"dy"^2 = (-1)/"2x"^2 = (-1)/(2 ("e"^(2"y"))^2)`

∴ `("d"^2y)/"dx"^2 xx ("d"^2x)/"dy"^2 = (- 2)/e^"2y" = - 2e^(-2"y")`