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(‌)We can simplify the argument of the logarithm:
‌=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
‌. Since
y=2loge(x)−2, we have:
‌=2⋅‌(loge(x))=2⋅‌=‌ Next, to find the second derivative
‌, we differentiate
‌=‌ again with respect to
x :
Comparing this result with the given options:
Option A:
−‌Option B:
−‌Option C:
−‌Option D:
‌We see that the correct answer is Option A:
−‌