Question: Find and simplify the difference quotient 
for f(x) = 
| Foundations:
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| 1) f(x + h) = ?
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| 2) How do you eliminate the 'h' in the denominator?
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| Answer:
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| 1)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x + h) = \frac{2}{3(x + h) + 1}}
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| 2) The numerator of the difference quotient is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}}
so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.
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Solution:
| Step 1:
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| The difference quotient that we want to simplify is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h} = \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h}
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| Step 2:
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| Now we simplify the numerator:
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\frac{f(x + h) - f(x)}{h}} &=& \displaystyle{\left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h}\\ & & \\ &=& \displaystyle{\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}} \end{array}}
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| Step 3:
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| Now we simplify the numerator:
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}} & = & \displaystyle{\frac{6x + 2 - 6x -6h -2}{h(3(x + h) + 1)(3x + 1))}}\\ & & \\ & = & \displaystyle{\frac{-6}{(3(x + h) + 1)(3x + 1))}} \end{array}}
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| Final Answer:
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-6}{(3(x + h) + 1)(3x + 1))}}
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