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Chapter 2: Q 4. (page 237)

Find the derivatives of $f (x) = \frac{1}{x^{2}} .$

Short Answer

Expert verified

The derivative of $f (x) = \frac{1}{x^{2}} i s - \frac{2}{x^{3}} .$

Step by step solution

01

Given Information

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

02

Simplification

$f' (x) = \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h} f' (\frac{1}{x^{2}}) = \lim_{h 鈫� 0} \frac{\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}}{h} = \lim_{h 鈫� 0} \frac{x^{2} - x^{2} - 2 h x - h^{2}}{x^{2} {(x + h)}^{2} h} = \lim_{h 鈫� 0} \frac{- (2 x + h) h}{{(x^{2} + h x)}^{2} h} = \lim_{h 鈫� 0} \frac{- 2 x - h}{{(x^{2} + h x)}^{2}} = \frac{- 2 x - 0}{{(x^{2} + h . 0)}^{2}} = \frac{- 2 x}{x^{4}} = - \frac{2}{x^{3}}$

Therefore, the derivate of $f (x) = \frac{1}{x^{2}} i s - \frac{2}{x^{3}} .$

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Most popular questions from this chapter

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.

27. $f (x) = 1 - x^{3}, x = - 1$

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

$f' (x) = (x^{4} - 8) (1 - 3 x^{5}); f (0) = 2$

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.

$f (x) = {(5 {(3 x^{4} - 1)}^{3} + 3 x - 1)}^{100}$

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

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

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