Question

Given that $f,g,h$are functions with values $f\left(2\right)=1,g\left(2\right)=-4,h\left(2\right)=3$and point -derivatives $f\text{'}\left(2\right)=3,g\text{'}\left(2\right)=0,h\text{'}\left(2\right)=-1$Calculate

$\left(\frac{f}{g}\right)\text{'}\left(2\right)$

Short answer

The answer is$\left(\frac{f}{g}\right)\text{'}\left(2\right)=-\frac{3}{4}$

Step-by-step solution

Given information

We are given that $f,g,h$are functions with values $f\left(2\right)=1,g\left(2\right)=-4,h\left(2\right)=3$and point -derivatives $f\text{'}\left(2\right)=3,g\text{'}\left(2\right)=0,h\text{'}\left(2\right)=-1$

Calculate

We are given $\left(\frac{f}{g}\right)\text{'}\left(2\right)$

Which can be simplified as

$$\begin{gathered} \frac{g\left(2\right)f\text{'}\left(2\right)-f\left(2\right)g\text{'}\left(2\right)}{{\left[g\right(2\left)\right]}^2} \\ =\frac{(-4)\left(3\right)-0}{16} \\ =-\frac{3}{4} \end{gathered}$$