Concept preview

What you will be able to do

1. 01Estimate the value of a derivative at a point by reading the slope of a tangent line drawn on a graph, using a clean rise-over-run triangle.
2. 02Estimate a derivative from a table of values using forward, backward, and symmetric difference quotients, and explain why the symmetric difference $\dfrac{f(a+h)-f(a-h)}{2h}$ is usually the most accurate.
3. 03Interpret the derivative as a FUNCTION — a rule that returns the slope of $f$ at every input — and connect the shape of the graph of $f$ to the graph of $f'$.

The derivative $f'(a)$ is the slope of the tangent line to $y=f(x)$ at the point where $x=a$. So if you are handed a graph, estimating $f'(a)$ is a geometry problem: draw the straight line that just grazes the curve at $x=a$, matching its direction there, and measure that line's slope. To measure a slope, pick two points that lie clearly on the tangent line — ideally where it crosses gridline intersections — and form a rise-over-run triangle. The slope is $\dfrac{\text{rise}}{\text{run}}=\dfrac{\Delta y}{\Delta x}$, the vertical change divided by the horizontal change between those two points.