How do you find derivative of f(x)=1x+2 using the definition of the derivative?

From the definition of derivative you have:f′(x)=limh→0f(x+h)−f(x)h&nbsp;where h is a small increment;and in our case:f′(x)=limh→01x+h+2−1x+2h==limh→01h(x+2−x−h−2(x+h+2)(x+2))==limh→01h(-h(x+h+2)(x+2))==−1(x+0+2)(x+2)=−1(x+2)2

"How do you find derivative of f(x)=1x+2 using..." was answered by our experts

High School
Calculus and Analysis
Calculus 1
Derivatives

Abbey Ellison

Answered question

2022-04-15

How do you find derivative of

f (x) = \frac{1}{x + 2}

using the definition of the derivative?

Answer & Explanation

superpms01wks1

Beginner2022-04-16Added 13 answers

From the definition of derivative you have:
$f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ where h is a small increment;
and in our case:
$f^{'} (x) = lim_{h \to 0} \frac{\frac{1}{x + h + 2} - \frac{1}{x + 2}}{h} =$
$= lim_{h \to 0} \frac{1}{h} (\frac{x + 2 - x - h - 2}{(x + h + 2) (x + 2)}) =$
$= \lim_{h \to 0} \frac{1}{h} (- \frac{h}{(x + h + 2) (x + 2)}) =$
$= - \frac{1}{(x + 0 + 2) (x + 2)} = - \frac{1}{{(x + 2)}^{2}}$

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