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 □.

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  □.

Accepted Answer

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

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QUESTION IMAGE

Question

Explanation:

Step1: Find $f(z)$

Step2: Substitute into the derivative formula

Step3: Expand the numerator

Step4: Simplify the limit

Step5: Evaluate the limit

Answer: