2026 Fall Section 3.2 The Derivatives as a Function
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Overview
Ara Araeipour develops the derivative as a function, moving from the secant-slope limit definition to notation, differentiability, tangent lines, and graphical interpretations. Worked examples use first principles to derive the tangent to √(3x + 1) at x = 8 and the derivatives of 5x + 1, 2/x, and 7x² + 4; graph sketches connect derivative values to slopes, increasing and decreasing intervals, and nondifferentiable points.
Key takeaways
- The derivative function f′(x) is obtained by allowing the point of tangency to vary, while f′(a) is the numerical tangent slope at one specified input.
- For f(x) = √(3x + 1), conjugate multiplication reduces the difference quotient to f′(x) = 3/[2√(3x + 1)], producing tangent line y = (3/10)x + 13/5 at x = 8.
- A function's increasing intervals correspond to positive derivative values, decreasing intervals to negative values, and smooth local extrema to derivative values of zero.
- A function is not differentiable at a discontinuity or sharp corner; vertical tangents and cusps can also prevent a finite derivative.
- For piecewise linear functions, the derivative is constant on each line segment and undefined at corners, which appear as holes in the derivative graph.
- Using the limit definition, the derivatives of 5x + 1, 2/x, and 7x² + 4 are respectively 5, −2/x², and 14x.
Chapters
- Average rate of change is the secant slope, expressed as [f(x) − f(a)]/(x − a) or [f(a + h) − f(a)]/h.
- The tangent slope is the limit of secant slopes as the second point approaches the first, giving f′(a).
- Replacing the fixed point a with a variable x defines f′(x) = lim as h → 0 of [f(x + h) − f(x)]/h.
- Derivative notation includes f′(x), y′, dy/dx, and the differentiation operator d/dx; differentiability on an interval requires a derivative at every point in it.
- Araeipour applies the difference quotient to f(x) = √(3x + 1), multiplying by the conjugate to resolve the 0/0 limit.
- After cancellation of h, the derivative function is f′(x) = 3/[2√(3x + 1)].
- At x = 8, f(8) = 5 and f′(8) = 3/10, so the tangent point is (8, 5) with slope 3/10.
- Point-slope form gives y − 5 = (3/10)(x − 8), or y = (3/10)x + 13/5.
- At a smooth local maximum or minimum, the horizontal tangent has slope zero, so f′ crosses or meets the x-axis at the same x-coordinate.
- Where f increases, f′ is positive; where f decreases, f′ is negative.
- A drawn tangent at x = 5 has an estimated slope of 3/2, giving an approximate point (5, 1.5) on the derivative graph.
- For a straight-line function, the derivative is constant: a line with slope 3 has derivative graph y = 3.
- Three common failures of differentiability are a sharp corner, a discontinuity, and a vertical tangent.
- A discontinuity rules out differentiability; a corner has unequal one-sided slopes and no single tangent slope.
- A vertical tangent is associated with derivative values tending to the same signed infinity from both sides.
- A vertical cusp differs from a vertical tangent: its one-sided derivative limits have opposite signs, such as −∞ from the left and +∞ from the right.
- To identify failures of differentiability on a graph, scan from left to right and classify features such as a discontinuity at x = 0, a vertical tangent, or a sharp corner.
- For a piecewise linear function, each segment produces a horizontal segment on f′ whose height equals the original segment's slope.
- The demonstrated segment slopes include −1/2, 1/2, 1, and −1, calculated from rise over run.
- At the three sharp corners, f′ is undefined, so the horizontal derivative pieces have holes at their corresponding x-values.
- For f(x) = 5x + 1, the difference quotient simplifies to 5h/h, giving f′(x) = 5.
- For f(x) = 2/x, combining terms with the common denominator x(x + h) cancels h and yields f′(x) = −2/x², for x ≠ 0.
- For f(x) = 7x² + 4, expanding 7(x + h)² and canceling matching terms leaves 14xh + 7h².
- Factoring and canceling h gives the limit of 14x + 7h as h → 0, so f′(x) = 14x.
Transcript excerpt
0:07 Good morning, class. We're going to chapter 3 derivatives, section 3.2, the derivative the derivatives as a function. So, we covered derivatives and rates of change, and now the derivative as a function. Reminding everyone, given y = f of x and point a, f of a, the slope of a secant line PQ, in other words, the average rate of change. If you have two points, you can calculate rise over run. Delta y over delta x, y sub 2 - y sub 1 over x sub 2 - x sub 1, or f of x - f of a over x - a, or in this format, where this is a and we move to the right by h, f of a + h - f of a
0:57 over h. They all mean the same thing. They are referring to the slope when two points are given. However, uh if we call this b and f of b, that works just as well. f of b - f of a over b - a. So, that all means the same thing. Now, the secant line, as we move Q towards P to create the red line, the 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. So, slope of a tangent line, also known as instantaneous rate of change, m of tangent line. If we look at this concept, limit of m of PQ as Q
1:46 approaches P, if we look at this one, the limit of f of x - f of a over x - a as x approaches a. If we look at this one, the limit of f of a + h minus f of a over h
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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.