Question

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

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

Answer 1

Answer: (D)

We have
$f^{\prime}(4)=5 ; g^{\prime}(4)=12 ; f(4) g(4)=2$
and $g(4)=6$
Now, on differentiating
$\left(\frac{f}{g}\right)^{\prime} =\frac{g f^{\prime}-f \cdot g^{\prime}}{g^{2}}$
$=\frac{(x) f^{\prime}(x) g-f(x) \cdot g^{\prime}(x)}{[g(x)]^{2}} $
$\therefore \left(\frac{f}{g}\right)(4)=\frac{6 \cdot 5-\frac{1}{3} \cdot 12}{6 \cdot 6}$
$\left[\because f(4) g(4)=2 \Rightarrow f(4)=\frac{2}{5}=\frac{1}{2}\right]$
$=\frac{30-4}{36}=\frac{26}{36}=\frac{13}{18} $