Practice: Higher-order derivatives

Self-check

Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.

1. What is the second derivative, in two sentences?

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A derivative is itself a function, so you can differentiate it again. The second derivative f'' is the derivative of the derivative: it measures how the slope is changing (the rate of change of f'), not the slope itself.

2. What is the second derivative of position, in physics?

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Acceleration. If s(t) is position, then s'(t) is velocity (how fast position changes) and s''(t) is acceleration (how fast velocity changes). Newton’s F = ma says force equals mass times this second derivative; classical mechanics is written in second derivatives. (The third derivative of position is jerk.)

3. What does the sign of f''(x) say about the graph’s shape?

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f''(x) > 0: the slope is increasing, so the graph cups upward (smiling). f''(x) < 0: the slope is decreasing, so the graph cups downward (frowning). f''(x) = 0 with a sign change is an inflection point, where the curvature flips between the two.

4. State the second-derivative test.

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At a critical point (f'(x) = 0): if f''(x) > 0, the graph cups upward there, so it is a local minimum (the cup holds water). If f''(x) < 0, it cups downward, so it is a local maximum. If f''(x) = 0, the test is inconclusive; look closer.

5. What is the oscillation equation, and where does it come from?

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f'' = -f: a function whose second derivative is its own negative. It comes from differentiating sine twice: sin' = cos, then cos' = -sin, so sin'' = -sin. The same holds for cosine. This equation describes anything whose acceleration pulls it back toward center in proportion to displacement, which is why sine and cosine govern springs, pendulums, sound, AC, and light.

6. What is special about every derivative of e^x?

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Every derivative of e^x, to any order, is e^x itself. The self-derivative property compounds: differentiate once, you get e^x; differentiate again, still e^x; and so on, forever. This relentless self-reproduction is what makes its Taylor series come out so clean in the next lesson.

Try it yourself, part 1: map the shape with the second-derivative test

Pen and paper, about 7 minutes. Take f(x) = x³ - 12x + 5. Find its critical points (where f' = 0), classify each as a local minimum or local maximum with the second-derivative test, find the function’s value there, and locate any inflection points.

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f(x)   = x³ - 12x + 5
f'(x)  = 3x² - 12 = 3(x² - 4)
f''(x) = 6x

Critical points (f'(x) = 0): 3(x² - 4) = 0, so x = 2 and x = -2.

Second-derivative test:

• At x = 2: f''(2) = 6·2 = 12 > 0, so this is a local minimum. Value: f(2) = 8 - 24 + 5 = -11.
• At x = -2: f''(-2) = 6·(-2) = -12 < 0, so this is a local maximum. Value: f(-2) = -8 + 24 + 5 = 21.

Inflection point: f''(x) = 6x = 0 at x = 0, and the curvature flips (negative for x < 0, positive for x > 0), so x = 0 is an inflection point, with f(0) = 5.

From two derivatives you have the full shape: a hill at (-2, 21), a valley at (2, -11), and a curvature flip at (0, 5), without plotting an extra point.

Try it yourself, part 2: a projectile

About 4 minutes. A ball is thrown upward from a 64-foot tower with initial velocity 48 ft/s, so its height is s(t) = -16t² + 48t + 64 (feet, after t seconds). Find: (1) the velocity, (2) the time at the top of the arc and the maximum height, (3) the constant acceleration (gravity), and (4) when it hits the ground and how fast it is moving then.

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Velocity is the first derivative:

s'(t) = -32t + 48

Top of the arc: velocity zero. -32t + 48 = 0, so t = 1.5 s. Maximum height:

s(1.5) = -16·(1.5)² + 48·(1.5) + 64 = -36 + 72 + 64 = 100 ft

Acceleration is the second derivative:

s''(t) = -32 ft/s²  (constant; gravity)

It is -32 everywhere, including at the peak where the velocity is momentarily zero. Gravity never lets up, even at the top of the arc.

Impact: s(t) = 0, so -16t² + 48t + 64 = 0, or t² - 3t - 4 = 0 = (t-4)(t+1), giving t = 4 s (rejecting t = -1). Velocity at impact: s'(4) = -32·4 + 48 = -80 ft/s (the negative sign means downward).

Three layers of motion from one polynomial: where it is, how fast, and how the speed is changing.

Flashcards

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Q. What is the second derivative?

A.

The derivative of the derivative: f'' measures how the slope f' is changing, not the slope itself. A function can have a large positive slope while that slope is shrinking (f' > 0 but f'' < 0).

Q. What is the physical meaning of the second derivative of position?

A.

Acceleration. s'(t) is velocity, s''(t) is how fast velocity changes. Newton’s F = ma says force = mass × this second derivative. The third derivative of position is jerk (the rate of change of acceleration).

Q. What does the sign of f''(x) say about the graph?

A.

f'' > 0: slope increasing, graph cups upward (smiling). f'' < 0: slope decreasing, graph cups downward (frowning). f'' = 0 with a sign change: inflection point, where curvature flips.

Q. State the second-derivative test.

A.

At a critical point (f' = 0): f'' > 0 means local minimum (cup holds water); f'' < 0 means local maximum; f'' = 0 is inconclusive. Anchor on the cup shape, not memorized signs.

Q. What is the oscillation equation, and where does it come from?

A.

f'' = -f. From sine: sin' = cos, cos' = -sin, so sin'' = -sin. Anything whose acceleration pulls it back toward center in proportion to displacement obeys this: springs, pendulums, sound, AC, light.

Q. What is special about every derivative of e^x?

A.

Every derivative of e^x, to any order, is e^x itself. The self-derivative property compounds: it stays e^x forever. This is why its Taylor series (next lesson) comes out so cleanly.

Q. Differentiate x⁴ repeatedly.

A.

x⁴ → 4x³ → 12x² → 24x → 24 → 0. Every polynomial eventually differentiates to zero, which is what the Taylor series exploits: a function is described by a finite tower of higher derivatives at a point.

Q. For f(x) = x³ - 3x, where are the local extrema and inflection point?

A.

f' = 3x² - 3, zero at x = ±1; f'' = 6x. At x = 1: f'' = 6 > 0, local minimum at (1, -2). At x = -1: f'' = -6 < 0, local maximum at (-1, 2). Inflection at x = 0 where f'' changes sign.

Q. Why does the second derivative matter in machine learning?

A.

It is curvature, which drives second-order optimization: Newton’s method uses the Hessian (matrix of second derivatives) to take better-informed steps than gradient descent. Adam keeps an informal curvature estimate; K-FAC approximates the Hessian. Loss-landscape analysis (saddle points, flat basins, sharp ridges) is second-derivative geometry.