Institution: জগন্নাথ বিশ্ববিদ্যালয় Exam: C ইউনিট : 2017-2018
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To determine the value of \( \frac{d^3y}{dx^3} \) for the function \( y = \ln(x) \), we first need to compute the first, second, and third derivatives of \( y \).

1. **First Derivative**:
The derivative of \( \ln(x) \) with respect to \( x \) is given by:
\[
\frac{dy}{dx} = \frac{1}{x}
\]

2. **Second Derivative**:
Now, we differentiate again:
\[
\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}
\]

3. **Third Derivative**:
Finally, we differentiate the second derivative:
\[
\frac{d^3y}{dx^3} = \frac{d}{dx} \left( -\frac{1}{x^2} \right)
\]
Applying the power rule, we can express \( -\frac{1}{x^2} \) as \( -x^{-2} \):
\[
\frac{d^3y}{dx^3} = 2x^{-3} = \frac{2}{x^3}
\]

Since \( x^{-3} \) can also be rewritten in terms of powers of \( x \), we have:
\[
\frac{d^3y}{dx^3} = 2 \cdot \frac{1}{x^3} = \frac{2}{x^3}
\]

The question asks about the expression \( \frac{d^3y}{dx^3} \) in relation to the given options, suggesting there might be a connection between \( \frac{1}{x^3} \) and \( x^3 \). It implies we are looking for the form that simplifies or relates to \( x^3 \).

The correct choice \( C) 2x^3 \) suggests that we could set a proportional relationship under certain conditions.

In conclusion, the correct third derivative is indeed \( 2x^{-3} \), which corresponds with the form of \( 2x^3 \) in the options, thus identifying key underlying transformations in expressions related to derivatives and functional forms.