The differential coefficient of ${x^6}$ with respect to ${x^3}$ is
A. $5{x^2}$
B. $3{x^3}$
C. $5{x^3}$
D. $2{x^3}$

Hint: In order to find the differential coefficient of one function with respect to another function, first derivative both the functions with respect to the common variable they are having. Then, divide the first value obtained by the second value, as we need the first function derivative with respect to the second function.
Complete step-by-step solution:
We are given two functions one is ${x^6}$ and another is ${x^3}$ and we need to find the coefficient of the first function with respect to another. For that first we need to find their differentiation separately.
Considering the first function ${x^6}$ to be $u$, which can be numerically written as:
$u = {x^6}$
Since, it is having the variable $x$, so differentiating $u$ with respect to $x$, and we get:
$\dfrac{{du}}{{dx}} = \dfrac{{d\left( {{x^6}} \right)}}{{dx}}$ ……(1)
From the differentiation rule, we know that:
$\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}$
So, comparing the above formula with the equation 1, we get:
$ \Rightarrow \dfrac{{du}}{{dx}} = 6{x^{6 - 1}} = 6{x^5}$ …..(2)
And, similarly considering the second function ${x^3}$ to be $v$, which is numerically written as:
$v = {x^3}$
Since, it is also having the variable $x$, so differentiating $v$ also with respect to $x$, and we get:
$\dfrac{{dv}}{{dx}} = \dfrac{{d\left( {{x^3}} \right)}}{{dx}}$ ……(3)
From the differentiation rule, we know that:
$\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}$
So, comparing the above formula with the equation 3, we get:
$ \Rightarrow \dfrac{{dv}}{{dx}} = 3{x^{3 - 1}} = 3{x^2}$ …..(4)
Since, we need to find the coefficient for the first function with respect to the second function, so basically, we need $\dfrac{{du}}{{dv}}$.
For that dividing equation 2 by equation 4, and we get:
$ \Rightarrow \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}} = \dfrac{{6{x^5}}}{{3{x^2}}}$
Which can be further written as:
$ \Rightarrow \dfrac{{du}}{{dv}} = 2{x^{5 - 2}}$
$ \Rightarrow \dfrac{{du}}{{dv}} = 2{x^3}$
Therefore, the differential coefficient of ${x^6}$ with respect to ${x^3}$ is $2{x^3}$.
Hence, Option 4 is correct.

Note: In the equation $\dfrac{{du}}{{dv}} = 2{x^{5 - 2}}$, we have used the law of radicals for the value $2{x^{5 - 2}}$, as according to that the power having same bases in division will be subtracted, for example: If $\dfrac{{{p^a}}}{{{p^b}}}$ have same base p, so their powers will be subtracted and written as ${p^{a - b}}$.