Question

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions having finite non-zero $3rd$ order derivatives $f\text{'}\text{'}\text{'}\left(x\right)$ and $g\text{'}\text{'}\text{'}\left(x\right)$ for all, $x\in R$. If $f\left(x\right)g\left(x\right)=1$ for all $x\in R$, then $\left(\frac{f\text{'}\text{'}\text{'}}{f\text{'}}\right)-\left(\frac{g\text{'}\text{'}\text{'}}{g\text{'}}\right)$.

• A. $3\left[\left(\frac{f\text{'}\text{'}}{g}\right)-\left(\frac{g\text{'}\text{'}}{f}\right)\right]$
• B. $3\left[\left(\frac{f\text{'}\text{'}}{f}\right)-\left(\frac{g\text{'}\text{'}}{g}\right)\right]$
• C. $3\left[\left(\frac{g\text{'}\text{'}}{g}\right)-\left(\frac{f\text{'}\text{'}}{g}\right)\right]$
• D. $3\left[\left(\frac{f\text{'}\text{'}}{f}\right)-\left(\frac{g\text{'}\text{'}}{f}\right)\right]$

Answer 1

The correct option is B

$3\left[\left(\frac{f\text{'}\text{'}}{f}\right)-\left(\frac{g\text{'}\text{'}}{g}\right)\right]$

Finding the value of $\left(\frac{f\text{'}\text{'}}{f\text{'}}\right)-\left(\frac{g\text{'}\text{'}}{g\text{'}}\right)$:

Given, $f\left(x\right)g\left(x\right)=1\dots \left(1\right)$

$$\begin{gathered} \Rightarrow g\left(x\right)=\frac{1}{f\left(x\right)} \\ \Rightarrow g\left(x\right)=f^{-1}\left(x\right) \end{gathered}$$

Differentiating $\left(1\right)$ with respect to $x$.

$\Rightarrow f\text{'}\left(x\right)g\left(x\right)+g\text{'}\left(x\right)f\left(x\right)=0$

Differentiating the above equation again with respect to $x$, we get

$\Rightarrow f\text{'}\text{'}\left(x\right)g\left(x\right)+f\text{'}\left(x\right)g\text{'}\left(x\right)+g\text{'}\text{'}\left(x\right)f\left(x\right)+g\text{'}\left(x\right)f\text{'}\left(x\right)=0$

Differentiating the above equation again with respect to $x$, we get

$$\begin{gathered} \Rightarrow f\text{'}\text{'}\text{'}\left(x\right)g\left(x\right)+g\text{'}\text{'}\text{'}\left(x\right)f\left(x\right)+3g\text{'}\left(x\right)f\text{'}\text{'}\left(x\right)+3f\text{'}\left(x\right)g\text{'}\text{'}\left(x\right)=0 \\ \Rightarrow \frac{f\text{'}\text{'}\text{'}\left(x\right)}{f\text{'}\left(x\right)}f\text{'}\left(x\right)g\left(x\right)+\frac{g\text{'}\text{'}\text{'}\left(x\right)}{g\text{'}\left(x\right)}g\text{'}\left(x\right)f\left(x\right)+3\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)}f\left(x\right)g\text{'}\left(x\right)+3\frac{g\text{'}\text{'}\left(x\right)}{g\left(x\right)}g\left(x\right)f\text{'}\left(x\right)=0 \\ \Rightarrow \left[\frac{f\text{'}\text{'}\text{'}\left(x\right)}{f\text{'}\left(x\right)}+3\frac{g\text{'}\text{'}\left(x\right)}{g\left(x\right)}\right]\left(f\text{'}\left(x\right)g\left(x\right)\right)=-\left[\frac{g\text{'}\text{'}\text{'}\left(x\right)}{g\text{'}\left(x\right)}+3\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)}\right]f\left(x\right)g\text{'}\left(x\right) \\ \Rightarrow -\left[\frac{f\text{'}\text{'}\text{'}\left(x\right)}{f\text{'}\left(x\right)}+3\frac{g\text{'}\text{'}\left(x\right)}{g\left(x\right)}\right]f\left(x\right)g\text{'}\left(x\right)=-\left[\frac{g\text{'}\text{'}\text{'}\left(x\right)}{g\text{'}\left(x\right)}+3\frac{f\text{'}\text{'}\left(x\right)}{g\left(x\right)}\right]f\left(x\right)g\text{'}\left(x\right) \end{gathered}$$

Hence, the value of $\left(\frac{f\text{'}\text{'}}{f\text{'}}\right)-\left(\frac{g\text{'}\text{'}}{g\text{'}}\right)$ is $3\left[\left(\frac{f\text{'}\text{'}}{f}\right)-\left(\frac{g\text{'}\text{'}}{g}\right)\right]$.

Thus, the correct answer is option (B).