Chapter 2: Q. 44 (page 184)

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises39-54
$f (x) = \frac{2}{1 - x}$

Short Answer

Expert verified

The value of $f' (x) = \frac{2}{{(1 - x)}^{2}}$

Step by step solution

Step 1. Given information

The given function $f (x) = \frac{2}{1 - x}$

Step 2. Finding the value of f'(x)

We know that $f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} . . . . . . . . . (1)$

Given $f (x) = \frac{2}{1 - x}$ then

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

Putting these values in(1)

$f' (x) = \lim_{h \to 0} \frac{\frac{2}{1 - x - h} - \frac{2}{1 - x}}{h} = \lim_{h \to 0} \frac{\frac{2 (1 - x) - 2 (1 - x - h)}{(1 - x - h) (1 - x)}}{h} = \lim_{h \to 0} \frac{\frac{2 - 2 x - 2 + 2 x + 2 h}{(1 - x - h) (1 - x)}}{h} = \lim_{h \to 0} \frac{\frac{2 h}{(1 - x - h) (1 - x)}}{h} = \lim_{h \to 0} \frac{2 h}{(1 - x - h) (1 - x)} \times \frac{1}{h} = \lim_{h \to 0} \frac{2}{(1 - x - h) (1 - x)}$

Put $h = 0$

$= \frac{2}{(1 - x - 0) (1 - x)} = \frac{2}{(1 - x) (1 - x)} = \frac{2}{{(1 - x)}^{2}}$

Hence,the value of $f' (x) = \frac{2}{{(1 - x)}^{2}}$

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Use the definition of the derivative to find $f'$ for each function $f$ in Exercises39-54
$f (x) = \frac{2}{1 - x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of f'(x)

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Use the definition of the derivative to find $f'$ for each function $f$ in Exercises39-54
$f (x) = \frac{2}{1 - x}$