Q. 14 - Calculus

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

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

Step-by-Step Solution

Answer

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

Step 1: 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 $f\text{'}\left(2\right)=3, g\text{'}\left(2\right)=0, h\text{'}\left(2\right)=-1$

Step 2: Calculate

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

Which can be simplified as

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