Question

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 - 9}.
the derivative of f(x)=\frac{4x}{x - 9} is □.

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

\[

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

\]

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

\[

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

\]

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:

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