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The difference quotient is simply how the average rate of change compares over an interval.
The difference quotient of [tex]\mathbf{f(x) = \frac 4x}[/tex] is [tex]\mathbf{-\frac{4}{x + h}}[/tex]
The function is given as:
[tex]\mathbf{f(x) = \frac 4x}[/tex]
The difference quotient for f(x) is calculated using:
[tex]\mathbf{Difference\ quotient = \frac{f(x + h) - f(x)}{h}}[/tex]
We have:
[tex]\mathbf{f(x) = \frac 4x}[/tex]
Substitute x + h for x
[tex]\mathbf{f(x + h) = \frac 4{x + h}}[/tex]
The difference quotient becomes
[tex]\mathbf{Difference\ quotient = \frac{f(x + h) - f(x)}{h}}[/tex]
[tex]\mathbf{Difference\ quotient = \frac{\frac{4}{x + h} - \frac{4}{x}}{h}}[/tex]
Take LCM
[tex]\mathbf{Difference\ quotient = \frac{\frac{4x - 4x - 4h}{x + h}}{h}}[/tex]
[tex]\mathbf{Difference\ quotient = \frac{\frac{- 4h}{x + h}}{h}}[/tex]
Rewrite as:
[tex]\mathbf{Difference\ quotient = \frac{- 4h}{x + h} \div h}[/tex]
Express as products
[tex]\mathbf{Difference\ quotient = \frac{- 4h}{x + h} \times \frac 1h}[/tex]
[tex]\mathbf{Difference\ quotient = \frac{- 4}{x + h}}[/tex]
[tex]\mathbf{Difference\ quotient = -\frac{4}{x + h}}[/tex]
Hence, the difference quotient of [tex]\mathbf{f(x) = \frac 4x}[/tex] is [tex]\mathbf{-\frac{4}{x + h}}[/tex]
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