difference quotient\ngiven the function $f(x)=\frac{3}{x}$, determine the difference quotient using the…

difference quotient\ngiven the function $f(x)=\frac{3}{x}$, determine the difference quotient using the following two - step process\n1. find $\frac{f(x + h)-f(x)}{h}$ without simplification\n2. find $\frac{f(x + h)-f(x)}{h}$ with simplification\n1. write the difference quotient for $f(x)=\frac{3}{x}$ without simplification\n$\frac{f(x + h)-f(x)}{h}=$\n1. write the difference quotient for $f(x)=\frac{3}{x}$ with simplification\n$\frac{f(x + h)-f(x)}{h}=$

difference quotient\ngiven the function $f(x)=\frac{3}{x}$, determine the difference quotient using the following two - step process\n1. find $\frac{f(x + h)-f(x)}{h}$ without simplification\n2. find $\frac{f(x + h)-f(x)}{h}$ with simplification\n1. write the difference quotient for $f(x)=\frac{3}{x}$ without simplification\n$\frac{f(x + h)-f(x)}{h}=$\n1. write the difference quotient for $f(x)=\frac{3}{x}$ with simplification\n$\frac{f(x + h)-f(x)}{h}=$

Answer

Explanation:

Step1: Find $f(x + h)$

Given $f(x)=\frac{3}{x}$, then $f(x + h)=\frac{3}{x + h}$.

Step2: Calculate the non - simplified difference quotient

Substitute $f(x + h)$ and $f(x)$ into $\frac{f(x + h)-f(x)}{h}$: $\frac{\frac{3}{x + h}-\frac{3}{x}}{h}$

Step3: Simplify the difference quotient

First, find a common denominator for the numerator: $\frac{\frac{3x-3(x + h)}{x(x + h)}}{h}=\frac{3x-3x-3h}{x(x + h)h}=\frac{-3h}{x(x + h)h}$ Then, cancel out the $h$ terms: $-\frac{3}{x(x + h)}$

Answer:

  1. $\frac{\frac{3}{x + h}-\frac{3}{x}}{h}$
  2. $-\frac{3}{x(x + h)}$