DERIVATIVES

The result

For f(x) = 3/x, the slope at x = 4 is -3/16. The point on the curve is (4, 3/4).

This overview prepares you to follow a short worked example, not a complete Calculus I lecture. Suggested prerequisites are function substitution, common denominators, distributing a negative sign, division by fractions, and introductory limits - the operations used in the tutorial. [V1 01:12] [V1 01:32] [V1 03:11] [V1 03:39]

Learning objectives are to construct the difference quotient for f(x) = 3/x at x = 4, simplify it, evaluate its limit, and interpret the result as a tangent slope. [V1 00:04] [V1 00:36] [V1 04:33]

The central takeaway is f′(4) = −3/16, whereas f(4) = 3/4: the first is a slope, the second a function value. [V1 01:25] [V1 04:25] Explanatory addition: cancellation takes place with h ≠ 0; only afterward is the limit as h approaches zero evaluated. [V1 03:50] [V1 04:13]

The generation used captions. A subsequent editorial spot-check confirmed the displayed final calculation at 4:34 and tangent graph at 4:50; this was not a frame-by-frame audit. [V1 04:39]

How to use this guide

Read the overview, follow the algebra, then attempt the three original practice problems before checking the separate answers. This guide covers one 5-minute tutorial, not a complete course or exam syllabus.

1. ELABORATION proceeds through the spoken calculation.

Step 1: identify the target as f′(4), not f(4), for f(x) = 3/x. [V1 00:04] [V1 00:08]

Step 2: write f′(4) = lim as h→0 of [f(4+h) − f(4)]/h. The entire difference is divided by h. [V1 00:36] [V1 00:42] [V1 00:45]

Step 3: substitute into the original function: f(4+h) = 3/(4+h) and f(4) = 3/4, giving lim as h→0 of [3/(4+h) − 3/4]/h. [V1 01:18] [V1 01:20] [V1 01:25]

Step 4: use the common denominator 4(4+h): 3/(4+h) − 3/4 = [12 − 3(4+h)]/[4(4+h)]. [V1 02:04] [V1 02:16] [V1 02:43] [V1 02:49]

Step 5: distribute the subtraction correctly: 12 − 3(4+h) = 12 − 12 − 3h = −3h. Thus the difference quotient is [−3h/[4(4+h)]]/h. [V1 03:11] [V1 03:31] [V1 03:36]

Step 6: division by h means multiplication by 1/h, so the quotient becomes −3h/[4(4+h)h] = −3/[4(4+h)]. [V1 03:39] [V1 03:50] [V1 04:04] Explanatory addition: these equalities require h ≠ 0 and h ≠ −4. For example, 0 < |h| < 1 satisfies both restrictions.

Cancellation does not make the original quotient defined at zero; it produces an expression with the same values nearby. This explains the tutorial's cancellation-before-limit sequence. [V1 03:50] [V1 04:13]

Step 7: the simplified denominator tends to 4·4 = 16, which is nonzero, so f′(4) = −3/16. This agrees with the spoken result; OpenStax snippets corroborate the general strategy of simplifying a complex fraction before applying limit laws, not this particular numerical example. [V1 04:18] [V1 04:24] [V1 04:25] [W3]

2. ALL THE DETAILS follow the source order. The instructor introduces f(x) = 3/x at x = 4 and identifies the requested quantity as f′(4). [V1 00:04] [V1 00:08]

Two approaches are offered: first obtain the derivative function and then substitute 4, or substitute 4 into the limit definition immediately. The tutorial chooses the second approach; it does not work out a general formula for f′(x). [V1 00:13] [V1 00:14] [V1 00:22] [V1 00:24] [V1 00:46]

The distinction matters: f′(4) is one number, while the derivative function assigns derivative values to inputs wherever the derivative exists. [W2] The point-specific limit is stated and repeated, then f(4+h) and f(4) are evaluated. [V1 00:36] [V1 00:48] [V1 01:07]

Fraction subtraction occupies the next stage: choose 4(4+h), multiply the first fraction by 4/4 and the second by (4+h)/(4+h), and combine their numerators. [V1 01:32] [V1 01:54] [V1 02:04] [V1 02:16] [V1 02:43] The source says that 4+h and 4 have only the common factor 1.

Read this as an informal symbolic common-denominator explanation, not a claim about the numerical factors of every possible substituted value of h; the calculation only needs 4(4+h) to be a valid common denominator. [V1 01:42] [V1 01:54]

The instructor then distributes −3, combines 12 − 12, changes division by h to multiplication by 1/h, cancels h, and evaluates the remaining limit. [V1 03:11] [V1 03:31] [V1 03:39] [V1 03:50] [V1 04:13] Finally, −3/16 is interpreted as the tangent slope at x = 4.

The instructor describes a blue graph of 3/x, the point (4, 3/4), and a red tangent line with that slope. The spoken description was subsequently checked against the displayed graph at 4:50; the exact result follows from the algebra, not from measuring the graph. [V1 04:33] [V1 04:42] [V1 04:50] [V1 04:53]

3. COMMON MISTAKES are instructional cautions added to the source, not errors attributed to the instructor. Do not replace f(4+h) by f(4)+h: substitution puts 4+h in the denominator, producing 3/(4+h). [V1 01:18] [V1 01:20]

Do not subtract fractions without a common denominator or forget the outer division by h. [V1 01:32] [V1 02:34] Do not turn −3(4+h) into −12+3h; both terms receive the negative multiplier. [V1 03:11]

Do not cancel h inside the sum 4+h: the source cancels the multiplicative factor h in −3h against the outer denominator h. [V1 03:31] [V1 03:50] Do not set h = 0 before simplifying: the original quotient would have numerator 3/4 −

3/4 and denominator 0, giving the undefined form 0/0, not the answer zero. This explanatory addition clarifies why the source simplifies before taking the limit. [V1 01:25] [V1 01:30] [V1 04:13]

Do not confuse the height 3/4 with the slope −3/16, or present the single number f′(4) as the whole derivative function. [V1 04:25] [V1 04:50] [W2] Finally, a described or displayed graph is an illustration, not an independent exact numerical proof of the slope.

The exact result here comes from the limit calculation. [V1 04:18] [V1 04:25] [V1 04:39]

4. ORIGINAL PRACTICE contains three newly written problems, not additional video content.

Problem 1, recall: for the tutorial's f(x) = 3/x, state f(4), f(4+h), and the limit defining f′(4). State the restrictions on h before cancellation. This revisits the source's substitution and difference quotient. [V1 00:36] [V1 01:20] [V1 01:25]

Problem 2, application: let g(x) = 5/x. Find g′(2) using only the limit definition, showing fraction subtraction, cancellation, and limit evaluation; then state the point of tangency and tangent slope. This is an original variation of the source method. [V1 01:32] [V1 03:39] [V1 04:33]

Problem 3, explanation: a student says, 'Because h approaches zero, set h = 0 immediately; the numerator becomes zero, so f′(4) = 0.' Explain the error, explain why cancellation is allowed, and distinguish f(4), f′(4), and f′(x).

This original diagnostic question targets the source's order of operations and its two proposed approaches. [V1 00:13] [V1 03:50] [V1 04:13]

5. ANSWERS give original, algebraically checked workings.

Problem 1: f(4) = 3/4 and f(4+h) = 3/(4+h). Therefore f′(4) = lim as h→0 of [3/(4+h) − 3/4]/h. Require h ≠ 0 because of the outer denominator, and h ≠ −4 because 4+h must be nonzero.

These values and the quotient reproduce the spoken setup; the explicit domain restrictions are explanatory additions. [V1 00:36] [V1 01:20] [V1 01:25]

Problem 2, original worked variation: g(2) = 5/2 and g(2+h) = 5/(2+h), so g′(2) = lim as h→0 of [5/(2+h) − 5/2]/h. Combine fractions: 5/(2+h) − 5/2 = [10 − 5(2+h)]/[2(2+h)] = [10 − 10 − 5h]/[2(2+h)] = −5h/[2(2+h)].

Dividing by h gives −5h/[2(2+h)h] = −5/[2(2+h)] for h ≠ 0, −2. Now take the limit: g′(2) = −5/[2·2] = −5/4. The point is (2, 5/2), and the tangent slope is −5/4.

Algebra check: multiplying −5/[2(2+h)] by h restores −5h/[2(2+h)], and splitting [10 − 5(2+h)]/[2(2+h)] restores 5/(2+h) − 5/2. Thus both the fraction subtraction and cancellation reverse correctly. This is new work using the tutorial's method, not a claim that the video contains this example. [V1 02:43] [V1 02:49] [V1 03:39] [V1 04:13]

Problem 3: immediate substitution gives (3/4 − 3/4)/0 = 0/0, which is undefined; it does not give zero. For nonzero h near zero, the quotient equals −3h/[4(4+h)h] = −3/[4(4+h)]. Taking the limit of this equal-nearby expression gives −3/16.

Check by multiplying the simplified expression by h: it returns −3h/[4(4+h)], the numerator difference found in the source. The cancellation and limit are separate operations: h is nonzero during the algebra, but its values approach zero in the limit. [V1 03:31] [V1 03:50] [V1 04:18]

Consequently, f(4) = 3/4 is the graph's height, f′(4) = −3/16 is its tangent slope there, and f′(x) denotes the derivative function rather than this one slope. [V1 04:25] [V1 04:50] [W2]

00:04 / The problem - f(x) = 3/x at x = 4.

00:36 / Set up the limit - The point-specific difference quotient.

01:20 / Substitute - Put 4+h into the denominator.

02:43 / Combine fractions - Use the common denominator 4(4+h).

03:50 / Cancel a factor - h is nonzero during simplification.

04:13 / Evaluate the limit - The simplified denominator tends to 16.

04:50 / Interpret the graph - The point and its tangent slope.

Source and adapted-content license

James Sousa / Mathispower4u, Use the Limit Definition of the Derivative to Find a Derivative Function Value. Changes: reorganized study notes, additional explanations and original practice. This adapted report is distributed under CC BY-SA 3.0 Unported. No endorsement is implied.

Original video

Publisher and license statement

CC BY-SA 3.0 Unported

What was checked

98 caption segments were available. The source arithmetic was independently derived; the final calculation and graph were spot-checked in the video. The original practice answer for g(x)=5/x at x=2 was independently verified as -5/4, with point (2,5/2). Full frame-by-frame visual coverage and student usefulness have not been tested.

Limits and additions

The explicit nonzero-h justification and denominator restrictions are not stated in the cancellation narration. They are supplied as explanatory additions, not attributed quotations. [V1 03:50] [V1 03:57]

The general derivative-function approach is mentioned but not worked out. This companion therefore does not invent a source derivation for f′(x). [V1 00:14] [V1 00:22] [V1 00:46]

Web validation was limited to returned search excerpts; full educational pages and the video's own licensing description were not inspected. The excerpts corroborate general concepts and publisher-level attribution, not unseen source content. [W1] [W2] [W3] [W9]

The general derivative formula referenced at 00:20 was not visually checked. Only equations recoverable from the spoken source are treated as source content. Web searches supplied excerpts, not full-document verification. Source captions end near 5:04 in the 5:09 video; no claim is made about unseen closing material.

Corroborated descriptions

Cross-video comparison is unavailable because only V1 was supplied. The source's difference-quotient method and tangent-slope interpretation agree with the corresponding OpenStax descriptions. [V1 00:36] [V1 04:33] [W1] [W2]

Qualifications

Qualifications and editorial updates

Editorial update: the final graph was visually spot-checked at 4:50 and agrees with the narrated function and tangent slope. A graph illustrates the result; the exact numerical value is established by the algebra. [V1 04:39] [V1 04:42] [V1 04:53]

No returned external snippet directly reproduces the full 3/x-at-4 calculation. Its result is checked here by explicit algebra, not presented as externally verified from search snippets. [V1 00:04] [V1 04:25]

The statement that 4 and 4+h have only the common factor 1 needs the symbolic-context qualification given above; it is not a universal numerical factor claim. [V1 01:42]

UNVERIFIED

The source asks for f′(4) when f(x) = 3/x.

Explicit in the supplied captions. No relevant returned snippet independently establishes the exact example; this status concerns external corroboration, not mathematical doubt.

[V1 00:04] [V1 00:08]

CORROBORATED

One may first find the derivative function and evaluate it at 4, or directly compute the derivative at 4; the tutorial chooses the latter.

The OpenStax snippet directly describes the derivative function as assigning derivative values to individual inputs. It supports the conceptual relationship, not verification of the video's unseen general formula.

[V1 00:14] [V1 00:22] [V1 00:24] [V1 00:46] [W2]

CORROBORATED

A derivative at a point is obtained as a limit of a difference quotient.

The OpenStax snippet explicitly supports the limit-of-difference-quotient definition. The exact formula [f(4+h) − f(4)]/h comes from the captions; mathematical notation is incompletely rendered in the snippet.

[V1 00:36] [V1 00:39] [V1 00:42] [V1 00:45] [W1]

UNVERIFIED

Substitution gives f(4+h) = 3/(4+h), f(4) = 3/4, and the quotient [3/(4+h) − 3/4]/h.

Directly supported by speech and checked by substitution. No returned external snippet contains these particular equalities.

[V1 01:18] [V1 01:20] [V1 01:25] [V1 01:30]

UNVERIFIED

The tutorial uses 4(4+h) as the common denominator and describes 4 and 4+h as having only the common factor 1.

The chosen common denominator is algebraically valid. The factor wording is informal and should not be generalized to numerical values after every possible substitution for h. No external snippet resolves the wording.

[V1 01:42] [V1 01:46] [V1 01:54]

UNVERIFIED

Multiplying the fractions by 4/4 and (4+h)/(4+h) gives the numerator difference [12 − 3(4+h)]/[4(4+h)].

Every component is spoken. Direct algebra confirms the equality when 4+h ≠ 0; no unseen board expression was needed.

[V1 02:04] [V1 02:16] [V1 02:43] [V1 02:49]

UNVERIFIED

Distributing −3 and combining constants changes 12 − 3(4+h) into −3h, while the outer denominator remains h.

Checked directly: 12 − 12 − 3h = −3h. External search snippets do not establish this source-specific step.

[V1 03:11] [V1 03:31] [V1 03:34] [V1 03:36]

CORROBORATED

Simplifying a complex fraction and canceling common factors before applying limit laws is a valid general strategy.

The OpenStax PDF snippet directly recommends simplification of complex fractions, cancellation of common factors, and subsequent application of limit laws. Only the returned excerpt was examined.

[V1 03:39] [V1 03:50] [V1 04:13] [W3]

UNVERIFIED

Here division by h and cancellation yield −3/[4(4+h)] for h ≠ 0 and h ≠ −4.

The exact cancellation is spoken and algebraically checked. The explicit restrictions are this companion's explanatory addition. The snippet supports the method generally but does not directly establish this expression or its restrictions.

[V1 03:41] [V1 03:45] [V1 03:50] [V1 04:04] [V1 04:10] [W3]

UNVERIFIED

The remaining limit is −3/[4·4] = −3/16, so f′(4) = −3/16.

Independently checked from the reconstructed algebra; the limiting denominator is 16, not zero. W4 supports rational-function limit evaluation as a topic but does not give the exact calculation.

[V1 04:13] [V1 04:18] [V1 04:24] [V1 04:25] [W4]

CORROBORATED

The derivative at a point represents the slope of the tangent line there.

Both OpenStax snippets directly support the tangent-slope interpretation. This corroborates the interpretation rather than independently verifying the specific numerical slope.

[V1 04:33] [V1 04:36] [W1] [W2]

UNVERIFIED

The instructor describes a blue graph of 3/x, the point (4, 3/4), and a red tangent with slope −3/16 as graphical verification.

During generation only narration was available. A later visual spot-check at 4:50 confirmed the blue curve, marked point and red tangent labeled -3/16. A graph alone does not establish an exact slope.

[V1 04:39] [V1 04:42] [V1 04:50] [V1 04:53] [V1 04:56]

CORROBORATED

Mathispower4u credits James Sousa, states CC BY-SA 3.0 Unported licensing, and lists the supplied tutorial title in its calculus collection.

Publisher snippets directly support authorship, site-level licensing, and the title listing; the Creative Commons snippet identifies the license. The timestamp anchors the associated tutorial, not a spoken attribution. The exact supplied YouTube URL was not independently resolved by search.

[W9] [W15] [W23]

W1 / 3.1 Defining the Derivative - Calculus Volume 1 | OpenStax

W2 / 3.2 The Derivative as a Function - Calculus Volume 1 | OpenStax

W3 / [PDF] Calculus Volume 1 - OpenStax

W4 / 2.3 The Limit Laws - Calculus Volume 1 | OpenStax

W9 / mathispower4u

W15 / mathispower4u

W19 / Legal Code - Attribution-ShareAlike 3.0 Unported - Creative Commons

W23 / Deed - Attribution-ShareAlike 3.0 Unported - Creative Commons