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Difference quotient for $$$f{\left(x \right)} = \frac{1}{x + 1}$$$
Your Input
Find the difference quotient for $$$f{\left(x \right)} = \frac{1}{x + 1}$$$.
Solution
The difference quotient is given by $$$\frac{f{\left(x + h \right)} - f{\left(x \right)}}{h}$$$.
To find $$$f{\left(x + h \right)}$$$, plug $$$x + h$$$ instead of $$$x$$$: $$$f{\left(x + h \right)} = \frac{1}{\left(x + h\right) + 1}$$$.
Finally, $$$\frac{f{\left(x + h \right)} - f{\left(x \right)}}{h} = \frac{\frac{1}{\left(x + h\right) + 1} - \frac{1}{x + 1}}{h} = - \frac{1}{\left(x + 1\right) \left(h + x + 1\right)}$$$.
Answer
The difference quotient for $$$f{\left(x \right)} = \frac{1}{x + 1}$$$A is $$$- \frac{1}{\left(x + 1\right) \left(h + x + 1\right)}$$$A.
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