Chapter 2: Q. 2 (page 196)

Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each function
$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$

Short Answer

Expert verified

$f' (x) = 4 x^{3} f' (x) = - \frac{2}{x^{3}} f' (x) = 3 x^{2} + 2 x f' (x) = \frac{x^{2} + 2 x}{{(x + 1)}^{2}}$

Step by step solution

Given Information.

Consider the given information.

$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$

Find the derivative of fx=x4.

Definition of derivative formula

$f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} f (x) = x^{4} f (x + h) = {(x + h)}^{4} = x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} f' (x) = \lim_{h \to 0} \frac{x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} - x^{4}}{h} f' (x) = \lim_{h \to 0} \frac{4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}}{h} f' (x) = \lim_{h \to 0} 4 x^{3} + 6 x^{2} h + 4 x h^{2} + h^{3} f' (x) = 4 x^{3}$

Find the derivative of fx=x-2.

Use the definition and find the derivative.

f (x) = x^{- 2} f (x + h) = {(x + h)}^{- 2} = \frac{1}{{(x + h)}^{2}} = \frac{1}{x^{2} + 2 x h + h^{2}} f' (x) = \lim_{h \to 0} \frac{\frac{1}{x^{2} + 2 x h + h^{2}} - \frac{1}{x^{2}}}{h} f' (x) = \lim_{h \to 0} \frac{x^{2} - (x^{2} + 2 x h + h^{2})}{h * x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = \lim_{h \to 0} \frac{x^{2} - x^{2} - 2 x h - h^{2}}{h * x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = \lim_{h \to 0} \frac{- 2 x - h}{x^{2} (x^{2} + 2 x h + h^{2})} f' (x) = - \frac{2}{x^{3}}

Find the derivative of fx=x2x+1.

Use the definition to find the derivative.

f' (x) = \lim_{h \to 0} [\frac{f (x + h) - f (x)}{h}] f' (x) = \lim_{h \to 0} (\frac{{(x + h)}^{2} (x + h + 1) - x^{2} (x + 1)}{h}) f' (x) = \lim_{h \to 0} (\frac{x^{3} + h^{2} + x^{2} + 3 x^{2} h + 3 x h^{2} + 2 x h + h^{3} - x^{3} - x^{2}}{h}) f' (x) = \lim_{h \to 0} (\frac{h^{3} + h^{2} + 3 x^{2} h + 3 x h^{2} + 2 x h}{h}) f' (x) = \lim_{h \to 0} h^{2} + h + 3 x^{2} + 3 x h + 2 x f' (x) = 3 x^{2} + 2 x

Find the derivative of fx=x2x+1.

Use the definition of derivative to find it.

f (x) = \frac{x^{2}}{x + 1} f' (x) = \lim_{h \to 0} [\frac{f (x + h) - f (x)}{h}] = \lim_{h \to 0} (\frac{\frac{{(x + h)}^{2}}{x + h + 1} - \frac{x^{2}}{x + 1}}{h}) = \lim_{h \to 0} (\frac{\frac{x h^{2} + x^{2} h + 2 x h + h^{2}}{(x + 1) (x + h + 1)}}{h}) = \lim_{h \to 0} (\frac{x h^{2} + x^{2} h + 2 x h + h^{2}}{(x + 1) (x + h + 1) h}) = \lim_{h \to 0} (\frac{x h + x^{2} + 2 x + h}{(x + 1) (x + h + 1)}) f' (x) = \frac{x^{2} + 2 x}{{(x + 1)}^{2}}

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each function
$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$

Short Answer

Step by step solution

Given Information.

Find the derivative of fx=x4.

Find the derivative of fx=x-2.

Find the derivative of fx=x2x+1.

Find the derivative of fx=x2x+1.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each functionf(x)=x4f(x)=x-2f(x)=x2(x+1)f(x)=x2x+1

Short Answer

Step by step solution

Given Information.

Find the derivative of fx=x4.

Find the derivative of fx=x-2.

Find the derivative of fx=x2x+1.

Find the derivative of fx=x2x+1.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each function
$f (x) = x^{4} f (x) = x^{- 2} f (x) = x^{2} (x + 1) f (x) = \frac{x^{2}}{x + 1}$