Higher-order derivatives describe how a function changes beyond its first rate of change. The first derivative tells slope or velocity, while the second derivative tells how that slope is changing. This makes higher-order derivatives important in motion, graphing, optimization, and modeling real systems.
They help connect the shape of a graph to measurable quantities like speed, acceleration, curvature, and stability.
To compute higher-order derivatives, take derivatives repeatedly in order. If f'(x) is the first derivative, then f''(x) is the derivative of f'(x), and f'''(x) is the derivative of f''(x). A positive second derivative means the graph is concave up, while a negative second derivative means the graph is concave down.
In physics, if position is s(t), then velocity is v(t) = s'(t) and acceleration is a(t) = s''(t).
Key Facts
- First derivative: f'(x) gives the slope of the tangent line and the instantaneous rate of change.
- Second derivative: f''(x) = d/dx[f'(x)] describes concavity and how the slope is changing.
- Third derivative: f'''(x) = d/dx[f''(x)] describes how the second derivative is changing.
- Leibniz notation: d2y/dx2 means the second derivative of y with respect to x.
- Motion formulas: v(t) = s'(t) and a(t) = v'(t) = s''(t).
- Concavity test: if f''(x) > 0, the graph is concave up; if f''(x) < 0, the graph is concave down.
Vocabulary
- Higher-order derivative
- A derivative found by differentiating a function more than once.
- Second derivative
- The derivative of the first derivative, often used to measure concavity or acceleration.
- Concavity
- The way a graph bends, either opening upward like a cup or downward like a cap.
- Inflection point
- A point where a graph changes concavity, usually where the second derivative changes sign.
- Acceleration
- The rate of change of velocity with respect to time, equal to the second derivative of position.
Common Mistakes to Avoid
- Treating f''(x) as the square of f'(x), which is wrong because f''(x) means differentiate f'(x), not multiply f'(x) by itself.
- Stopping after one derivative when asked for the second derivative, which gives slope but not concavity or acceleration.
- Assuming f''(x) = 0 always means an inflection point, which is wrong because the second derivative must change sign there.
- Mixing up positive slope with concave up, which is wrong because f'(x) describes whether the function increases or decreases, while f''(x) describes how the slope changes.
Practice Questions
- 1 Find f'(x), f''(x), and f'''(x) for f(x) = 2x4 - 5x3 + 3x - 7.
- 2 A particle has position s(t) = t3 - 6t2 + 9t meters. Find its velocity and acceleration at t = 4 seconds.
- 3 A graph has f'(x) > 0 and f''(x) < 0 on an interval. Describe whether the function is increasing or decreasing, and whether its slope is getting larger or smaller.