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Chapter 2: Q. 2 (page 196)

Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each function
$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$

Short Answer

Expert verified

$f' (x) = 4 x^{3} f' (x) = - \frac{2}{x^{3}} f' (x) = 3 x^{2} + 2 x f' (x) = \frac{x^{2} + 2 x}{{(x + 1)}^{2}}$

Step by step solution

01

Given Information.

Consider the given information.

$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$

02

Find the derivative of fx=x4.

Definition of derivative formula

$f' (x) = \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h} f (x) = x^{4} f (x + h) = {(x + h)}^{4} = x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} f' (x) = \lim_{h 鈫� 0} \frac{x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} - x^{4}}{h} f' (x) = \lim_{h 鈫� 0} \frac{4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}}{h} f' (x) = \lim_{h 鈫� 0} 4 x^{3} + 6 x^{2} h + 4 x h^{2} + h^{3} f' (x) = 4 x^{3}$

03

Find the derivative of fx=x-2.

Use the definition and find the derivative.

f (x) = x^{- 2} f (x + h) = {(x + h)}^{- 2} = \frac{1}{{(x + h)}^{2}} = \frac{1}{x^{2} + 2 x h + h^{2}} f' (x) = \lim_{h 鈫� 0} \frac{\frac{1}{x^{2} + 2 x h + h^{2}} - \frac{1}{x^{2}}}{h} f' (x) = \lim_{h 鈫� 0} \frac{x^{2} - (x^{2} + 2 x h + h^{2})}{h * x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = \lim_{h 鈫� 0} \frac{x^{2} - x^{2} - 2 x h - h^{2}}{h * x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = \lim_{h 鈫� 0} \frac{- 2 x - h}{x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = - \frac{2}{x^{3}}

04

Find the derivative of fx=x2x+1.

Use the definition to find the derivative.

f' (x) = \lim_{h 鈫� 0} [\frac{f (x + h) 鈭� f (x)}{h}] f' (x) = \lim_{h 鈫� 鈥� 0} (\frac{{(x + h)}^{2} (x + h + 1) - x^{2} (x + 1)}{h}) f' (x) = \lim_{h 鈫� 鈥� 0} (\frac{x^{3} + h^{2} + x^{2} + 3 x^{2} h + 3 x h^{2} + 2 x h + h^{3} - x^{3} - x^{2}}{h}) f' (x) = \lim_{h 鈫� 鈥� 0} (\frac{h^{3} + h^{2} + 3 x^{2} h + 3 x h^{2} + 2 x h}{h}) f' (x) = \lim_{h 鈫� 鈥� 0} h^{2} + h + 3 x^{2} + 3 x h + 2 x f' (x) = 3 x^{2} + 2 x

05

Find the derivative of fx=x2x+1.

Use the definition of derivative to find it.

f (x) = \frac{x^{2}}{x + 1} f' (x) = \lim_{h 鈫� 0} [\frac{f (x + h) 鈭� f (x)}{h}] = \lim_{h 鈫� 鈥� 0} (\frac{\frac{{(x + h)}^{2}}{x + h + 1} - \frac{x^{2}}{x + 1}}{h}) = \lim_{h 鈫� 鈥� 0} (\frac{\frac{x h^{2} + x^{2} h + 2 x h + h^{2}}{(x + 1) (x + h + 1)}}{h}) = \lim_{h 鈫� 鈥� 0} (\frac{x h^{2} + x^{2} h + 2 x h + h^{2}}{(x + 1) (x + h + 1) h}) = \lim_{h 鈫� 鈥� 0} (\frac{x h + x^{2} + 2 x + h}{(x + 1) (x + h + 1)}) f' (x) = \frac{x^{2} + 2 x}{{(x + 1)}^{2}}

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Most popular questions from this chapter

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

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Sketch a graph of the associated slope function f' for each function f.

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Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

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