Derivative at a point (Calc 1; Lecture 1-6; Fall 26)

Overview

Beard Meets Calculus develops the derivative at a point from secant-line limits, showing that it measures instantaneous rate of change and gives the slope of the best local linear approximation. The lecture defines the derivative using both a variable point and an increment h, tests the definition on |x| and 2x² + 3x + 7, and uses point-slope form to find tangent lines, including y = 4x for sin(4x) at x = 0.

Key takeaways

The derivative f′(a) is the limit of secant slopes as their two points approach one another, making it the instantaneous rate of change at a.
The equivalent derivative definitions using b → a and h → 0 describe the same limit; the h form often simplifies algebra because cancellation can target h directly.
For f(x) = |x|, the difference quotient at zero has one-sided values 1 and −1, so the derivative does not exist at the graph’s kink.
Applying the difference quotient to 2x² + 3x + 7 yields f′(a) = 4a + 3 after expanding, canceling constant-in-h terms, and removing a factor of h.
A tangent line at x = a has equation y = f(a) + f′(a)(x − a), combining the function value and derivative into a local linear approximation.
The derivative of sin(4x) at zero is 4, since sin(4h)/h = 4·sin(4h)/(4h) approaches 4; its tangent line is y = 4x.

Chapters

0:00 Quickfire Review: Fraction Simplification and a Sine Limit

The opening arithmetic review gives 1 for the alternating powers expression.
Combining the displayed fractions produces -6h/(5 + 3h).
Using sin(u) ≈ u near zero, sin(5t)/sin(3t) approaches 5/3 as t approaches zero.

1:52 Asymptote Notation and an Indeterminate Limit at Negative Infinity

Horizontal asymptotes are written as y = constant; vertical asymptotes are written as x = constant.
The limit of √(x² + 8x + 21) − √(x² − 6x + 14) as x approaches −∞ has the indeterminate form ∞ − ∞.
Because both radicals grow large, comparing their dominant behavior requires additional algebra rather than direct substitution.

6:15 Rationalizing the Radical Difference with a Conjugate

Multiplying by the conjugate, √(x² + 8x + 21) + √(x² − 6x + 14), converts the numerator using a difference of squares.
The numerator simplifies to 14x + 7, leaving a ratio with a sum of square roots in the denominator.
As x approaches −∞, the numerator tends to −∞ and the denominator to +∞, so the expression still needs to be scaled.

11:10 Scaling by 1/x and Preserving the Sign at Negative Infinity

Dividing numerator and denominator by x exposes the terms that vanish, including 7/x, 8/x, 21/x², −6/x, and 14/x².
Inside the radicals, dividing by x² leaves each radicand approaching 1.
Since x is negative near −∞, √(x²) = |x| = −x; retaining this sign gives the limit −14/2 = −7.

18:36 From Secant Lines to Two Definitions of the Derivative

The derivative f′(a) is defined as limᵦ→ₐ [f(b) − f(a)]/(b − a), the limit of average rates of change as the two points meet.
Setting b = a + h gives the equivalent form f′(a) = limₕ→₀ [f(a + h) − f(a)]/h, which often makes cancellation easier.
The derivative is the instantaneous or local rate of change and supplies the slope of the tangent line that best approximates the function near a.

26:04 Why |x| Is Not Differentiable at the Origin

The graph of f(x) = |x| remains V-shaped under any zoom at x = 0, so it does not locally resemble one line.
The difference quotient at zero is |h|/h: it equals 1 for h > 0 and −1 for h < 0.
Because the one-sided limits disagree, the limit does not exist and f′(0) is undefined.

31:21 Deriving the Polynomial Rule from the Difference Quotient

For f(x) = 2x² + 3x + 7, substitute a + h everywhere x appears in f(a + h), using parentheses to avoid substitution errors.
Expanding and subtracting f(a) cancels the terms without h; the remaining numerator is 4ah + 2h² + 3h.
After factoring and canceling h, the limit as h approaches zero gives f′(a) = 4a + 3.

40:02 Reading Tangent-Line Information from Point-Slope Form

At x = a, the tangent line passes through (a, f(a)) and has slope f′(a).
Its equation is y = f(a) + f′(a)(x − a), which records both the function value and its local rate of change.
If 3x − 7 is tangent to y = f(x) at x = 4, then f(4) = 5 and f′(4) = 3.

44:06 Finding the Tangent to sin(4x) at the Origin

For f(x) = sin(4x), the point at x = 0 is (0, 0).
The derivative quotient becomes limₕ→₀ sin(4h)/h; rewriting it as 4 · sin(4h)/(4h) gives slope 4.
The tangent line through (0, 0) with slope 4 is y = 4x, matching the local approximation sin(4x) ≈ 4x.

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