Question

the derivative as a function
question
completed: 0 of 6 | my score: 0/6 pts (0%
use the formula f(x)=\lim_{z\to x}\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.
f(x)=\frac{3}{x + 10}
f(x)=\square

Explanation:

Step1: Find f(z)

Given \(f(x)=\frac{3}{x + 10}\), then \(f(z)=\frac{3}{z + 10}\).

Step2: Substitute f(z) and f(x) into the derivative formula

\[

$$\begin{align*} f^{\prime}(x)&=\lim_{z ightarrow x}\frac{f(z)-f(x)}{z - x}\\ &=\lim_{z ightarrow x}\frac{\frac{3}{z + 10}-\frac{3}{x + 10}}{z - x}\\ &=\lim_{z ightarrow x}\frac{\frac{3(x + 10)-3(z + 10)}{(z + 10)(x + 10)}}{z - x}\\ &=\lim_{z ightarrow x}\frac{3x+30 - 3z-30}{(z - x)(z + 10)(x + 10)}\\ &=\lim_{z ightarrow x}\frac{3x-3z}{(z - x)(z + 10)(x + 10)}\\ &=\lim_{z ightarrow x}\frac{-3(z - x)}{(z - x)(z + 10)(x + 10)} \end{align*}$$

\]

Step3: Simplify the limit

Cancel out the \((z - x)\) terms:
\[

$$\begin{align*} f^{\prime}(x)&=\lim_{z ightarrow x}\frac{-3}{(z + 10)(x + 10)}\\ &=\frac{-3}{(x + 10)(x + 10)}\\ &=-\frac{3}{(x + 10)^2} \end{align*}$$

\]

Answer:

\(-\frac{3}{(x + 10)^2}\)