Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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3:06 minutes

Problem 3.2.26

Textbook Question

Using the Alternative Formula for Derivatives

Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.

g(x) = 1 + √x

Verified step by step guidance

Identify the function g(x) = 1 + √x and the point at which you want to find the derivative, which is x.

Substitute g(x) into the alternative formula for derivatives: g'(x) = lim (z → x) [(g(z) − g(x)) / (z − x)].

Express g(z) in terms of z: g(z) = 1 + √z. Substitute this into the formula: g'(x) = lim (z → x) [(1 + √z − (1 + √x)) / (z − x)].

Simplify the expression inside the limit: g'(x) = lim (z → x) [(√z − √x) / (z − x)].

To evaluate the limit, multiply the numerator and the denominator by the conjugate of the numerator: (√z + √x). This will help eliminate the square roots and simplify the expression for taking the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental concept in calculus. It is the limit of the average rate of change of the function over an interval as the interval approaches zero. Understanding derivatives is crucial for analyzing the behavior of functions, such as finding slopes of tangent lines and rates of change.

Limit

A limit is a fundamental concept in calculus that describes the value a function approaches as the input approaches a certain point. Limits are essential for defining derivatives and integrals, and they help in understanding the behavior of functions at points where they may not be explicitly defined. Calculating limits involves evaluating the function's behavior as the variable approaches a specific value.

Alternative Formula for Derivatives

The alternative formula for derivatives, f'(x) = lim (z → x) (f(z) − f(x)) / (z − x), provides a way to calculate the derivative using limits. This formula emphasizes the concept of instantaneous rate of change by considering the difference quotient as z approaches x. It is particularly useful for functions where the standard derivative rules are not easily applicable, requiring a deeper understanding of limits and function behavior.

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Related Practice

Textbook Question

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

f(x) = x² − 3x + 4

Textbook Question

[Technology Exercise]

Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph

y = (√(x + h) − √x)/h

for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

Textbook Question

Derivative of y = |x| Graph the derivative of f(x) = |x|. Then graph y = (|x| − 0)/(x − 0) = |x|/x. What can you conclude?

Textbook Question

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

g(x) = x / (x − 1)

Textbook Question

Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT / (V − nb)) − (an² / V²),

in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)

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Textbook Question

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km / q) + cm + (hq / 2),

where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

Find dA/dq and d²A/dq².

Textbook Question

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.

d. During what year did home prices drop most rapidly and what is an estimate of this rate?

Textbook Question

Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.

b. During what days does the population seem to be increasing fastest? Slowest?

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Using the Alternative Formula for DerivativesUse the formulaf'(x) = lim (z → x) (f(z) − f(x)) / (z − x)to find the derivative of the functions in Exercises 23–26.g(x) = 1 + √x

Key Concepts

Derivative

Limit

Alternative Formula for Derivatives

Watch next

Using the Alternative Formula for Derivatives

Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.

g(x) = 1 + √x