To find the derivative of the given function [tex]\( f(x) = \frac{4x}{7x - 18} \)[/tex] using the Quotient Rule, we should follow these steps:
1. Identify the numerator and the denominator functions:
- [tex]\( u(x) = 4x \)[/tex]
- [tex]\( v(x) = 7x - 18 \)[/tex]
2. Recall the Quotient Rule:
The Quotient Rule states that if you have a function [tex]\( \frac{u(x)}{v(x)} \)[/tex], its derivative is given by:
[tex]\[
\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}
\][/tex]
where [tex]\( u' \)[/tex] is the derivative of [tex]\( u(x) \)[/tex] and [tex]\( v' \)[/tex] is the derivative of [tex]\( v(x) \)[/tex].
3. Calculate the derivatives of the numerator and the denominator:
- [tex]\( u'(x) = \frac{d}{dx} [4x] = 4 \)[/tex]
- [tex]\( v'(x) = \frac{d}{dx} [7x - 18] = 7 \)[/tex]
4. Apply the Quotient Rule:
[tex]\[
f'(x) = \frac{(4) \cdot (7x - 18) - (4x) \cdot (7)}{(7x - 18)^2}
\][/tex]
5. Simplify the numerator:
[tex]\[
(4)(7x - 18) - (4x)(7) = 28x - 72 - 28x = -72
\][/tex]
6. Plug the simplified numerator back into the fraction:
[tex]\[
f'(x) = \frac{-72}{(7x - 18)^2}
\][/tex]
So, the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = \frac{4x}{7x - 18} \)[/tex] using the Quotient Rule is:
[tex]\[
-\frac{72}{(7x - 18)^2}
\][/tex]
The correct answer is:
[tex]\[
-\frac{72}{(7x - 18)^2}
\][/tex]
