use the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of f(x)=\\frac{4x}{x…

Explanation:

Step1: Find $f(z)$

Given $f(x)=\frac{4x}{x - 9}$, then $f(z)=\frac{4z}{z - 9}$.

Step2: Substitute into the derivative formula

[ \begin{align*} f^{\prime}(x)&=\lim_{z\rightarrow x}\frac{\frac{4z}{z - 9}-\frac{4x}{x - 9}}{z - x}\ &=\lim_{z\rightarrow x}\frac{\frac{4z(x - 9)-4x(z - 9)}{(z - 9)(x - 9)}}{z - x}\ &=\lim_{z\rightarrow x}\frac{4z(x - 9)-4x(z - 9)}{(z - x)(z - 9)(x - 9)} \end{align*} ]

Step3: Expand the numerator

[ \begin{align*} 4z(x - 9)-4x(z - 9)&=4zx-36z-4xz + 36x\ &=36x-36z\ &=- 36(z - x) \end{align*} ]

Step4: Simplify the limit

[ \begin{align*} f^{\prime}(x)&=\lim_{z\rightarrow x}\frac{-36(z - x)}{(z - x)(z - 9)(x - 9)}\ &=\lim_{z\rightarrow x}\frac{-36}{(z - 9)(x - 9)} \end{align*} ]

Step5: Evaluate the limit

Substitute $z = x$ into $\frac{-36}{(z - 9)(x - 9)}$, we get $f^{\prime}(x)=\frac{-36}{(x - 9)^2}$.

Answer:

$\frac{-36}{(x - 9)^2}$