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Chapter 2: Q 2. (page 237)

Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h 鈫� 0$ definition of the derivative and (b) the $z 鈫� x$ definition of the derivative.
$f (x) = x^{3}$

Short Answer

Expert verified

$f' (x) = 3 x^{2}$

Step by step solution

01

Given Information

The given expression is $f (x) = x^{3}$

02

Simplification

Using (a)

$f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{f (x + h) - f (x)}{h}$

$f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{{(x + h)}^{3} - x^{3}}{h}$

role="math" localid="1650970063541" $f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{x^{3} + h^{3} + 3 x^{2} h + 3 x h^{2} - x^{3}}{h}$

role="math" localid="1650970073601" $f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{h (h^{2} + 3 x^{2} + 3 x h)}{h}$

role="math" localid="1650970082470" $f^{'} (x) = \underset{h 鈫� 0}{l i m} h^{2} + 3 x^{2} + 2 x h$

Applying limit

$f (x) = 2 x^{2}$

Using (b)

role="math" localid="1650970131635" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{f (z) - f (x)}{z - x}$

role="math" localid="1650969972120" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{z^{3} - x^{3}}{z - x}$

role="math" localid="1650969988908" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{(z - x) (z^{2} + x^{2} + 2 z x)}{z - x}$

$f^{'} (x) = \underset{z 鈫� x}{l i m} (z^{2} + x^{2} + 2 z x)$

Applying limit

$f^{'} (x) = 3 x^{2}$

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Most popular questions from this chapter

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

$a) f (x) = {(3 x + 1)}^{4} b) f (x) = {(\frac{1}{x} + \frac{1}{x^{2}})}^{2} c) f (x) = {(x + 1)}^{2} \sqrt{x} d) f (x) = \frac{{(x + 1)}^{2}}{\sqrt{x}}$

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.

$f (x) = {(1 - 4 x)}^{2} {(3 x^{2} + 1)}^{9}$

Velocity $v (t)$ is the derivative of position $s (t)$ . It is also true that acceleration $a (t)$ (the rate of change of velocity) is the derivative of velocity. If a race car鈥檚 position in miles t hours after the start of a race is given by the function $s (t)$ , what are the units of $s (1.2)$ ? What are the units and real-world interpretation of $v (1.2)$ ? What are the units and real-world interpretations of $a (1.2)$ ?

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

$f' (x) = {(3 x + 1)}^{3}, f (2) = 1$

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.

$f (x) = \sqrt{x^{2} + 1}$

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