Schneider Lang
05/05/2024 · Middle School
Simplify the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the given function. \( f(x)=\frac{12}{x} \)
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Step-by-step Solution
Calculate or simplify the expression \( (12/(x+h)-12/x)/h \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(\frac{12}{\left(x+h\right)}-\frac{12}{x}\right)}{h}\)
- step1: Remove the parentheses:
\(\frac{\frac{12}{x+h}-\frac{12}{x}}{h}\)
- step2: Subtract the terms:
\(\frac{-\frac{12h}{x\left(x+h\right)}}{h}\)
- step3: Multiply by the reciprocal:
\(-\frac{12h}{x\left(x+h\right)}\times \frac{1}{h}\)
- step4: Reduce the fraction:
\(-\frac{12}{x\left(x+h\right)}\)
- step5: Multiply the terms:
\(-\frac{12}{x^{2}+hx}\)
The simplified form of the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x)=\frac{12}{x} \) is \( -\frac{12}{x^{2}+hx} \).
Quick Answer
The simplified difference quotient is \( -\frac{12}{x^{2}+hx} \).
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