Chapter 2: Q 1. (page 237)

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

Short Answer

Expert verified

$f' (x) = 2 x$

Step by step solution

Given Information

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

Simplification

Using (a)

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

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

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

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

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

Applying limit

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

Using (b)

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

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

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

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

Applying limit

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

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

Short Answer

Step by step solution

Given Information

Simplification

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Find the derivatives of the functions that follow using (a) the h→0definition of thederivative and (b) the z→xdefinition of the derivative.fx=x2

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

Logic and Functions

Discrete Mathematics

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Pure Maths

Study anywhere. Anytime. Across all devices.

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