30 秒でわかる
Grant Sanderson explains the second derivative through changing slope, finite differences and acceleration, then introduces jerk and the connection to Taylor series.
- 1SlopeThe second derivative measures how the first derivative changes.
- 2CurvatureIts sign describes whether slope increases or decreases.
- 3Notationd²f/dx² means differentiating twice with respect to x.
- 4Small stepsComparing two small increments gives an intuition for the second derivative.
- 5AccelerationAcceleration is the second derivative of position.
- 6JerkThe third derivative of position is jerk.
ポイント
- The second derivative measures how the first derivative changes.
- Its sign describes whether slope increases or decreases.
- d²f/dx² means differentiating twice with respect to x.
- Comparing two small increments gives an intuition for the second derivative.
- Acceleration is the second derivative of position.
- The third derivative of position is jerk.
チャプターごとに整えた全文を読む
Graphs and second derivatives
How the slope changes.
In this short calculus lesson, I introduce higher-order derivatives as preparation for Taylor series, focusing on the second derivative and its interpretation in graphs and motion.
The first derivative of a function gives the slope of its graph. The second derivative is the derivative of that slope: it measures how the first derivative changes as the input changes.
Where the graph bends upward, its slope increases and the second derivative is positive; where it bends downward, the slope decreases and the second derivative is negative. The size indicates how quickly the slope changes.
Notation and small steps
A finite-difference intuition.
The notation d²f/dx² abbreviates differentiating f twice with respect to x. It should not be read as ordinary multiplication of a variable d.
To build intuition, take two equal small steps of size dx and compare the function’s changes over them. The difference between the changes is typically proportional to dx² for a sufficiently smooth function.
Dividing that difference by dx² and taking the limit as dx approaches zero gives the second derivative, when the relevant limit exists.
Motion and higher derivatives
Acceleration, jerk and approximation.
For a position function of time, the first derivative is velocity and the second is acceleration. Positive or negative acceleration describes direction; whether an object speeds up also depends on the sign of its velocity.
The third derivative of position is called jerk: it describes how acceleration changes. Higher derivatives also provide information used to approximate functions in Taylor series.
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