2026 Fall Section 3.3 Basic Differentiation Formulas

Overview

Section 3.3 develops the basic differentiation formulas from the difference quotient, then applies the constant, power, constant-multiple, sum, and difference rules to polynomial, negative-exponent, and radical expressions. It also explains why the derivative of e^x is e^x and uses derivatives to find tangent lines, horizontal tangencies, and velocity and acceleration from position functions.

Key takeaways

The derivative at a point is the limit of the difference quotient, and that value supplies the slope in the tangent-line equation y - f(a) = f'(a)(x - a).
The power rule, d/dx(x^n) = n x^(n - 1), follows from the binomial theorem because all terms beyond n x^(n - 1) vanish in the limit after division by h.
Rewriting expressions as powers makes the basic rules broadly useful: 1/x^5 becomes x^(-5), and an nth root of x^m becomes x^(m/n).
The exponential base e is defined so that lim as h approaches 0 of (e^h - 1)/h equals 1, which is why d/dx(e^x) = e^x.
For s(t) = 2t^2 - 5t + 4, the velocity is 4t - 5 and acceleration is constant at 4 m/s²; the object reaches 31 m/s after 9 seconds.
Horizontal tangent points are found by solving f'(x) = 0, then substituting each x-value into the original function to recover the coordinate.

Chapters

0:00 From Secant Slopes to the Derivative Function

The difference quotient can be written as a change in y over a change in x, [f(x) - f(a)]/(x - a), or [f(a + h) - f(a)]/h.
A tangent line is the limiting position of secant lines as the second point approaches the first.
The derivative at a is the limiting tangent slope, and the tangent-line equation follows from point-slope form: y - f(a) = f'(a)(x - a).
The derivative function generalizes this limit to each x: f'(x) = lim as h approaches 0 of [f(x + h) - f(x)]/h.

2:30 Constant Functions, Compound Interest, and Core Differentiation Rules

A constant function f(x) = c has derivative 0 because its graph is horizontal and its difference quotient is 0.
The number e is introduced through the limit (1 + 1/n)^n as n approaches infinity, equivalent to lim as h approaches 0 of (1 + h)^(1/h).
The power rule states d/dx(x^n) = n x^(n - 1), while a constant multiple can be carried outside the derivative.
The sum and difference rules differentiate each term separately; the section also lists derivatives of sine, cosine, tangent, secant, cosecant, and cotangent.

7:49 Deriving the Power Rule with the Binomial Theorem

The proof starts from the derivative definition applied to f(x) = x^n and expands (x + h)^n.
The binomial theorem identifies x^n as the term that cancels with the subtracted x^n, while the next term is n x^(n - 1)h.
After dividing by h, every remaining term contains a positive power of h and approaches 0 as h approaches 0.
The surviving term establishes f'(x) = n x^(n - 1); factorials and combinations are reviewed to explain the binomial coefficients.

12:20 Applying the Power Rule to Polynomials and Rewritten Expressions

Examples apply the power rule to powers such as x^6 and t^4, and show that differentiating a linear function kx gives the constant k.
For a polynomial, the derivative is found term by term, lowering each exponent by 1 and multiplying by its original exponent.
Expressions that are not immediately in power-rule form are rewritten first, such as replacing a square root with x^(1/2).
The examples emphasize simplifying with exponent rules before differentiating, then presenting the result in a conventional form.

18:30 Differentiating Negative Powers and Radical Expressions

The reciprocal 1/x^5 is rewritten as x^(-5), differentiated to -5x^(-6), then returned to fraction form as -5/x^6.
A quotient of powers such as x^5/x^9 is simplified to x^(-4) before applying the power rule.
Radicals are converted to fractional exponents, including an nth root of x^m as x^(m/n), so the power rule applies directly.
After differentiation, negative or fractional powers can be converted back to denominator or radical notation to match the original expression.

27:14 Tangent Lines and Horizontal Tangencies from Derivatives

For a curve with derivative 2x - 3, the slope at x = 4 is 5; the point on the curve is (4, 3), giving tangent line y = 5x - 17.
A horizontal tangent requires f'(x) = 0 because a horizontal line has slope zero.
For f(x) = x^3 - 3x^2 + 5, f'(x) = 3x^2 - 6x, so horizontal tangencies occur at x = 0 and x = 2.
Substitution into the original function gives the corresponding points (0, 5) and (2, 1).

30:28 Why e^x Differentiates to Itself, Plus Higher Derivatives

For f(x) = a^x, the derivative definition factors out a^x and leaves the limit (a^h - 1)/h.
The base e is characterized by lim as h approaches 0 of (e^h - 1)/h = 1, which makes the derivative of e^x equal to e^x.
The lecture also presents e as the exponential base whose graph has slope 1 at the y-intercept (0, 1).
Higher derivatives are found by differentiating again; for e^x - x, the first derivative is e^x - 1 and the second derivative is e^x.

36:57 Trigonometric and Mixed-Function Derivative Practice

For y = sin t + π cos t, the derivative is cos t - π sin t.
A mixed example combines a fourth root, an exponential term, and a cosine term, using fractional exponents and the basic trig rules.
The derivative of cos t contributes -sin t, so differentiating a negative cosine term can produce a positive sine term.
Equivalent answer forms are shown, including fractional powers versus radicals and a coefficient multiplying sine.

40:57 Position, Average Velocity, Instantaneous Velocity, and Acceleration

For position s(t) = 2t^2 - 5t + 4 meters, average velocity from 2 to 4 seconds is [s(4) - s(2)]/(4 - 2) = 7 m/s.
Instantaneous velocity is the derivative of position: v(t) = s'(t) = 4t - 5, so v(6) = 19 m/s.
Acceleration is the second derivative, a(t) = v'(t) = s''(t) = 4 m/s²; differentiating velocity divides meters per second by seconds.
Setting 4t - 5 = 31 solves for the time when the object reaches 31 m/s: t = 9 seconds.

Transcript excerpt

0:11 Good morning class. Here's chapter 3, derivatives. Section 3.3, basic differentiation formulas. So we covered derivatives and rates of change. The derivative as a function. And now basic differentiation formulas. Reminding you that again a seeant line where we have two points on a curve. We can calculate the slope rise over run and there are different ways of writing delta y over delta x y sub 2 - y sub 1 / x sub 2 - x sub1. If you go with this one f of x - f of a x - 8. If you go with this one f of a plus h - f of a over a plus h - a which is h. They're all referring to the same thing. uh B f of B becomes F of B minus F of A B minus A. There is no difference and

1:01 they're all the same. Normally we call this the difference quotient. If we have several seeant lines and we want to bring Q closer to P in order to create a tangent line. The tangent line is the limiting position of the secant line PQ as Q approaches P. A tangent line is a line that touches a curve at a specific point without crossing it. slope of a tangent line and that slope if we stick with this as q approaches p limit of m of pq as q approaches p with this one the limit of f of x - f of x - a as x approaches a if we go with this one the limit of f of a plus h - f of a h as h approaches and that is the slope or frime of a or

1:53 the derivative at a. Now you want to come up with the equation of a tangent

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.