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Chapter 2: Q 1. (page 237)

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

Short Answer

Expert verified

$f' (x) = 2 x$

Step by step solution

01

Given Information

The given expression is $f (x) = x^{2}$

02

Simplification

Using (a)

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

$f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{{(x + h)}^{2} - x^{2}}{h}$

role="math" localid="1650969133678" $f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{x^{2} + h^{2} + 2 x h - x^{2}}{h}$

$f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{h (h + 2 x)}{h}$

$f^{'} (x) = \underset{h 鈫� 0}{l i m} h + 2 x$

Applying limit

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

Using (b)

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

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

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

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

Applying limit

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

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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.

$f (x) = \frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}}$

Use the definition of the derivative to find $f$ for each function $f$ in Exercises $39 - 54$

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

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the $z 鈫� x$ definition of the derivative

b) by using the $h 鈫� 0$ definition of the derivative

use the definition of the derivative to prove the quotient rule

Use (a) the $h 鈫� 0$ definition of the derivative and then

(b) the $z 鈫� c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23鈥�38.

25. $f (x) = \frac{1}{x}, x = - 1$

See all solutions

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