Question

The differential coefficient of ${\text{x}}^6$ with respect to ${\mathrm{x}}^3$ is

• A. $5{\mathrm{x}}^2$
• B. $3{\mathrm{x}}^3$
• C. $5{\mathrm{x}}^3$
• D. $2{\mathrm{x}}^3$

Answer 1

The correct option is D

$2{\mathrm{x}}^3$

Explanation for the correct option:

Let

$$\begin{gathered} \mathrm{u}={\mathrm{x}}^6,\frac{\mathrm{du}}{\mathrm{dx}}=6{\mathrm{x}}^5 \\ \mathrm{v}={\mathrm{x}}^3,\frac{\mathrm{dv}}{\mathrm{dx}}=3{\mathrm{x}}^2 \end{gathered}$$

Now,

$\begin{array}{rcl}\frac{\mathrm{du}}{\mathrm{dv}}& =& \frac{\left({\displaystyle \frac{\mathrm{du}}{\mathrm{dx}}}\right)}{\left({\displaystyle \frac{\mathrm{dv}}{\mathrm{dx}}}\right)}\\ & =& \frac{6{\mathrm{x}}^5}{3{\mathrm{x}}^2}\\ & =& 2{\mathrm{x}}^3\end{array}$

Hence, the correct answer is option (D).