To find the value of [tex]\(\left( \frac{f}{g} \right)'(9)\)[/tex], we will use the quotient rule for differentiation. The quotient rule states that if you have two differentiable functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], then the derivative of their quotient [tex]\(\frac{f(x)}{g(x)}\)[/tex] is given by:
[tex]\[
\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}
\][/tex]
We are given the following values:
- [tex]\(f(9) = -6\)[/tex]
- [tex]\(f'(9) = 3\)[/tex]
- [tex]\(g(9) = -8\)[/tex]
- [tex]\(g'(9) = 7\)[/tex]
We need to find [tex]\(\left( \frac{f}{g} \right)'(9)\)[/tex].
Substitute [tex]\(x = 9\)[/tex] into the quotient rule formula:
[tex]\[
\left( \frac{f}{g} \right)'(9) = \frac{f'(9) \cdot g(9) - f(9) \cdot g'(9)}{g(9)^2}
\][/tex]
Substitute the given values into the formula:
[tex]\[
\left( \frac{f}{g} \right)'(9) = \frac{3 \cdot (-8) - (-6) \cdot 7}{(-8)^2}
\][/tex]
Calculate the numerator:
[tex]\[
3 \cdot (-8) = -24
\][/tex]
[tex]\[
-6 \cdot 7 = -42
\][/tex]
[tex]\[
\text{Numerator} = -24 - (-42) = -24 + 42 = 18
\][/tex]
Calculate the denominator:
[tex]\[
(-8)^2 = 64
\][/tex]
Now, combine the numerator and the denominator:
[tex]\[
\left( \frac{f}{g} \right)'(9) = \frac{18}{64}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{18}{64} = 0.28125
\][/tex]
Thus, the value of [tex]\(\left( \frac{f}{g} \right)'(9)\)[/tex] is:
[tex]\[
0.28125
\][/tex]
