Chapter 2: Q. 46 (page 184)

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59
$f (x) = \frac{1}{x^{3}}$

Short Answer

Expert verified

The value of $f' (x) = \frac{- 3}{x^{4}}$

Step by step solution

Step 1. Given information

The given function $f (x) = \frac{1}{x^{3}}$

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 $f (x) = \frac{1}{x^{3}}$

Then $f (x + h) = \frac{1}{{(x + h)}^{3}}$

Putting there values in (1)

$f' (x) = \lim_{h \to 0} \frac{\frac{1}{{(x + h)}^{3}} - \frac{1}{x^{3}}}{h} = \lim_{h \to 0} \frac{\frac{x^{3} - {(x + h)}^{3}}{{(x + h)}^{3} x^{3}}}{h} = \lim_{h \to 0} \frac{\frac{x^{3} - (x^{3} + h^{3} + 3 x^{2} h + 3 x h^{2})}{x^{3} {(x + h)}^{3}}}{h} = \lim_{h \to 0} \frac{\frac{x^{3} - x^{3} - h^{3} - 3 x^{3} h - 3 x h^{2}}{x^{3} {(x + h)}^{3}}}{h} = \lim_{h \to 0} \frac{\frac{- h^{3} - 3 x^{2} h - 3 x h^{2}}{x^{3} {(x + h)}^{3}}}{h} = \lim_{h \to 0} \frac{\frac{h (- h^{2} - 3 x^{2} - 3 x h)}{x^{3} {(x + h)}^{3}}}{h} = \lim_{h \to 0} \frac{- h^{2} - 3 x^{2} - 3 x h}{x^{3} {(x + h)}^{3}}$

Putting $h = 0$

$= \frac{- {(0)}^{2} - 3 x^{2} - 3 x (0)}{x^{3} {(x + 0)}^{3}} = \frac{0 - 3 x^{2} - 0}{x^{3} (x^{3})} = \frac{- 3 x^{2}}{x^{6}} = \frac{- 3}{x^{4}}$

Hence,the value of $f' (x) = \frac{- 3}{x^{4}}$

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Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59
$f (x) = \frac{1}{x^{3}}$

Short Answer

Step by step solution

Step 1. Given information

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

One App. One Place for Learning.

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Use the definition of the derivative to find f'for each function fin Exercises 34-59f(x)=1x3

Short Answer

Step by step solution

Step 1. Given information

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

One App. One Place for Learning.

Most popular questions from this chapter

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Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59
$f (x) = \frac{1}{x^{3}}$