Question

If $f^{\prime }\left(4\right)=5,g^{\prime }\left(4\right)=12,f\left(4\right)g\left(4\right)=2$ and $g\left(4\right)=6$, then
$\left(\frac{f}{g}\right)\left(4\right)$ is

• A. $\frac{5}{36}$
• B. $\frac{11}{18}$
• C. $\frac{23}{36}$
• D. $\frac{13}{18}$
• E. $\frac{19}{36}$

Answer 1

Answer: (D)

$\frac{13}{18}$

We have

$f^{\prime }\left(4\right)=5;g^{\prime }\left(4\right)=12;f\left(4\right)g\left(4\right)=2$

Now on differentiating

${\left(\frac{f}{g}\right)}^{{}^{\prime }}=\frac{g{f}^{\prime }-f{g}^{\prime }}{g^2}=\frac{x{f}^{\prime }\left(x\right)g-f\left(x\right)g^{\prime }\left(x\right)}{{\left(g\right(x\left)\right)}^2}$

$\therefore \left(\frac{f}{g}\right)\left(4\right)=\frac{6.5-\frac{1}{3}.12}{6.6}[\because f(4\left)g\right(4)=2\Rightarrow f(4)=\frac{2}{6}=\frac{1}{3}]$

$=\frac{13}{18}$