Chapter 2: Q 3. (page 237)

Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h \to 0$ definition of the derivative and (b) the $z \to x$ definition of the derivative.
$f (x) = \frac{1}{x}$

Short Answer

Expert verified

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

Step by step solution

Given Information

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

Simplification

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

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

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

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

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

Applying limit

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

Using (b)

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

$f^{'} (x) = \underset{z \to x}{l i m} \frac{\frac{1}{z} - \frac{1}{x}}{z - x}$

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

$f^{'} (x) = \underset{z \to x}{l i m} - \frac{1}{z x}$

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

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Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h \to 0$ definition of the derivative and (b) the $z \to x$ definition of the derivative.
$f (x) = \frac{1}{x}$

Short Answer

Step by step solution

Given Information

Simplification

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Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the h→0definition of the derivative and (b) the z→xdefinition of the derivative.fx=1x

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Logic and Functions

Discrete Mathematics

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h \to 0$ definition of the derivative and (b) the $z \to x$ definition of the derivative.
$f (x) = \frac{1}{x}$