8A F11 Q12

Question: Find and simplify the difference quotient Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {f(x+h)-f(x)}{h}}} for f(x) = ${\frac {2}{3x+1}}$

Foundations
1) f(x + h) = ?
2) How do you eliminate the 'h' in the denominator?
Answer:
1) Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x+h)={\frac {2}{3(x+h)+1}}} the difference quotient is a difference of fractions divided by h.
2) The numerator is ${\frac {2}{3(x+h)+1}}-{\frac {2}{3x+1}}$ so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.

Solution:

Step 1:
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}

Step 2:
Now we simplify the numerator:
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} \frac{f(x + h) - f(x)}{h} &=& \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h\\ &=& \frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))} \end{array}}

Arithmetic:
Now we simplify the numerator:
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} \frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))} & = & \frac{6x + 2 - 6x -6h -2}{h(3(x + h) + 1)(3x + 1))}\\ & = & \frac{-6}{h(3(x + h) + 1)(3x + 1))} \end{array}}

Final Answer:
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}{h(3(x + h) + 1)(3x + 1))}}

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