To find the second derivative of the given function, let's first simplify and differentiate the expression step-by-step. The function is: y=loge(‌
x2
e2
) We can simplify the argument of the logarithm: ‌
x2
e2
=x2e−2 Now, using the property of logarithms that states loge(ab)=loge(a)+loge(b), we get: y=loge(x2e−2)=loge(x2)+loge(e−2) Since loge(e−2)=−2, we can further simplify: y=loge(x2)+loge(e−2)=loge(x2)−2=2loge(x)−2 Now, let's find the first derivative ‌
dy
dx
. Since y=2loge(x)−2, we have: ‌
dy
dx
=2⋅‌
d
dx
(loge(x))=2⋅‌
1
x
=‌
2
x
Next, to find the second derivative ‌
d2y
dx2
, we differentiate ‌
dy
dx
=‌
2
x
again with respect to x :
‌
d2y
dx2
=‌
d
dx
(‌
2
x
)=2⋅‌
d
dx
(x−1)=2⋅(−1)x−2=−‌
2
x2
Comparing this result with the given options: Option A: −‌
2
x2
Option B: −‌
1
x
Option C: −‌
1
x2
Option D: ‌
2
x2
We see that the correct answer is Option A: −‌