A derivative is not just one slope at one point, but a new function that gives the slope of the original function at every x-value where the slope exists. If f(x) describes position, height, cost, or temperature, then f'(x) describes how fast that quantity is changing. Thinking of the derivative as a function helps connect a graph's shape to a graph of its rates of change.
This idea is central to optimization, motion, modeling, and interpreting real data.
Key Facts
- Derivative definition: f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
- If f is increasing at x, then f'(x) > 0.
- If f is decreasing at x, then f'(x) < 0.
- If f has a horizontal tangent at x, then f'(x) = 0.
- For f(x) = x^n, d/dx[x^n] = n x^(n - 1).
- The derivative graph f'(x) plots x-values from f on the horizontal axis and tangent slopes on the vertical axis.
Vocabulary
- Derivative
- The derivative of a function is the instantaneous rate of change of the function with respect to its input.
- Derivative function
- A derivative function assigns to each input x the slope of the tangent line to the original function at that x-value.
- Tangent line
- A tangent line is a line that touches a curve at a point and has the same instantaneous direction as the curve there.
- Slope
- Slope measures vertical change divided by horizontal change, often written as rise over run.
- Critical point
- A critical point is a point in the domain of a function where f'(x) = 0 or where f'(x) does not exist.
Common Mistakes to Avoid
- Treating f'(x) as the height of f(x). This is wrong because f'(x) gives the slope of f at x, not the y-value of the original function.
- Assuming f'(x) is positive whenever f(x) is above the x-axis. This is wrong because the sign of f'(x) depends on whether f is rising or falling, not whether f is positive or negative.
- Marking f'(x) = 0 at every point where f(x) crosses the x-axis. This is wrong because f'(x) = 0 occurs where the tangent line is horizontal.
- Forgetting that sharp corners can make the derivative undefined. This is wrong because a derivative requires a single well-defined tangent slope at the point.
Practice Questions
- 1 For f(x) = x^2 - 4x + 1, find f'(x) and compute f'(3).
- 2 A function has tangent slopes of 2, 0, -3, and 1 at x = -1, 0, 2, and 4. Plot these four points on the graph of f'(x).
- 3 Suppose the graph of f rises until x = 1, has a horizontal tangent at x = 1, then falls until x = 5. Describe the sign of f'(x) on the intervals x < 1, at x = 1, and 1 < x < 5.