Question

the function f is defined as f(x)=\frac{7}{2x^{2}-4x}.
find f(x + 6).
write your answer without parentheses, and simplify it as much as possible.
f(x + 6)=

Explanation:

Step1: Find the derivative of \(f(x)\)

Use the quotient rule \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Here \(u = 7\), \(u^\prime=0\), \(v = 2x^{2}-4x\), \(v^\prime = 4x - 4\).

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Step2: Substitute \(x\) with \(x + 6\)

Replace \(x\) in \(f^\prime(x)\) with \(x+6\).
For the numerator: \(-28(x + 6)+28=-28x-168 + 28=-28x-140\)
For the denominator: \([2(x + 6)^{2}-4(x + 6)]^{2}\)
First, expand \(2(x + 6)^{2}-4(x + 6)\):

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So \(f^\prime(x + 6)=\frac{-28x-140}{(2x^{2}+20x + 48)^{2}}\)
Factor numerator and denominator:
Numerator: \(-28x-140=-28(x + 5)\)
Denominator: \(2x^{2}+20x + 48=2(x^{2}+10x + 24)=2(x + 4)(x+6)\), \((2x^{2}+20x + 48)^{2}=4(x + 4)^{2}(x + 6)^{2}\)

$$ f^\prime(x + 6)=\frac{-28(x + 5)}{4(x + 4)^{2}(x + 6)^{2}}=\frac{-7(x + 5)}{(x + 4)^{2}(x + 6)^{2}} $$

Answer:

\(\frac{-7(x + 5)}{(x + 4)^{2}(x + 6)^{2}}\)