Question

Given $F\left(x\right)=f\left(x\right)\varnothing \left(x\right)$ and$f\text{'}\left(x\right)\varnothing \text{'}\left(x\right)=c$ then $\frac{F\text{'}\text{'}\left(x\right)}{F\left(x\right)}-\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)}-\frac{\varnothing \text{'}\text{'}\left(x\right)}{\varnothing \left(x\right)}=?$

• A. $\frac{c}{f\left(x\right)\varnothing \left(x\right)}$
• B. $\frac{2c}{f\left(x\right)\varnothing \left(x\right)}$
• C. $\frac{3c}{f\left(x\right)\varnothing \left(x\right)}$
• D. None of these

Answer 1

The correct option is B

$\frac{2c}{f\left(x\right)\varnothing \left(x\right)}$

Explanation for the correct option:

Find the value of $\frac{F\text{'}\text{'}\left(x\right)}{F\left(x\right)}-\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)}-\frac{\varnothing \text{'}\text{'}\left(x\right)}{\varnothing \left(x\right)}$:

Given $F\left(x\right)=f\left(x\right)\varnothing \left(x\right)$ and$f\text{'}\left(x\right)\varnothing \text{'}\left(x\right)=c$

we have $F\left(x\right)=f\left(x\right)\varnothing \left(x\right)$

Differentiate with respect to x, we get

$$\begin{gathered} F\text{'}\left(x\right)=f\text{'}\left(x\right)\varnothing \left(x\right)+f\left(x\right)\varnothing \text{'}\left(x\right) \\ F\text{'}\text{'}\left(x\right)=f\text{'}\left(x\right)\varnothing \text{'}\left(x\right)+f\text{'}\text{'}\left(x\right)\varnothing \left(x\right)+f\text{'}\left(x\right)\varnothing \text{'}\left(x\right)+f\left(x\right)\varnothing \text{'}\text{'}\left(x\right) \\ =2f\text{'}\left(x\right)\varnothing \text{'}\left(x\right)+f\left(x\right)\varnothing \text{'}\text{'}\left(x\right)+f\text{'}\text{'}\left(x\right)\varnothing \left(x\right) \\ \Rightarrow \frac{F\text{'}\text{'}\left(x\right)}{f\left(x\right)\varnothing \left(x\right)}=\frac{2c+f\left(x\right)\varnothing \text{'}\text{'}\left(x\right)+f\text{'}\text{'}\left(x\right)\varnothing \left(x\right)}{f\left(x\right)\varnothing \left(x\right)} \\ \Rightarrow \frac{F\text{'}\text{'}\left(x\right)}{F\left(x\right)}=\frac{2c}{f\left(x\right)\varnothing \left(x\right)}+\frac{\varnothing \text{'}\text{'}\left(x\right)}{\varnothing \left(x\right)}+\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)} \\ \Rightarrow \frac{F\text{'}\text{'}\left(x\right)}{F\left(x\right)}-\frac{f\text{'}\text{'}\left(x\right)}{f\left(x\right)}-\frac{\varnothing \text{'}\text{'}\left(x\right)}{\varnothing \left(x\right)}=\frac{2c}{f\left(x\right)\varnothing \left(x\right)} \end{gathered}$$

Hence, the correct option is (B).