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Chapter 2: Q 4. (page 237)

$h 鈫� 0$ 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.
$f (x) = \frac{1}{x^{2}}$

Short Answer

Expert verified

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

Step by step solution

01

Given Information

The given expression is $f (x) = \frac{1}{x^{2}}$

02

Simplification

Using (a)

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

$f^{'} (x) = \underset{h 鈫� 0}{l i m} \frac{\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}}{h}$

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

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

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

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

Applying limit

role="math" localid="1650971983306" $f' (x) = - \frac{2 x}{x^{4}}$

role="math" localid="1650971992157" $f^{'} (x) = - \frac{2}{x^{3}}$

Using (b)

localid="1650972003166" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{f (z) - f (x)}{z - x}$

role="math" localid="1650972080100" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{\frac{1}{z^{2}} - \frac{1}{x^{2}}}{z - x}$

role="math" localid="1650972101501" $f^{'} (x) = \underset{z 鈫� x}{l i m} \frac{\frac{x^{2} - z^{2}}{z^{2} x^{2}}}{z - x}$

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

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

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

Applying limit

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

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

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Most popular questions from this chapter

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

$f' (x) = x (4 - 2 x); f (0) = 0$

For each function graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

For each function $f (x)$ and interval $[a, b]$ in Exercises $81 - 86$ , use the Intermediate Value Theorem to argue that the function must have at least one real root on $[a, b]$ . Then apply Newton鈥檚 method to approximate that root.

localid="1648369345806" $f (x) = x^{4} - 2, [a, b] = [1, 2]$ .

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.

26. $f (x) = \frac{1}{x^{2}}, x = 2$

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

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

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