Derivatives: from average change to slope at a point
Before you start: Read a function graph, calculate a line's slope, and expand simple brackets.
Begin with an average rate of change
For y = f(x), the average rate between x = a and x = a + h is [f(a + h) − f(a)]/h, where h ≠ 0. This is the slope of the secant line joining two graph points. If x is time in seconds and f(x) is distance in metres, its units are metres per second. For a track-height graph, the slope describes steepness instead; it is not automatically speed.
Worked example: the slope of f(x) = x² at x = 2
At x = 2, f(2) = 4. At x = 2 + h, f(2 + h) = (2 + h)² = 4 + 4h + h². Subtract the starting height and divide by h: [(4 + 4h + h²) − 4]/h = (4h + h²)/h = 4 + h, for h ≠ 0.
For h = 0.1, this secant slope is 4.1; for h = −0.1, it is 3.9. As h approaches zero from either side, the slopes approach 4. That limiting value is f′(2), the derivative at 2. We simplify before taking the limit; substituting h = 0 into the original quotient would incorrectly ask us to divide 0 by 0.
Use rules after understanding the limit
For positive integer n, the power rule gives the derivative of xⁿ as n xn−1. Constants have derivative zero, and sums can be differentiated term by term. For f(x) = 3x² − 2x + 5, f′(x) = 6x − 2. At x = 2, the slope is 10. Multiply the original coefficient by the exponent; do not simply reduce the power.
What positive, negative and zero slopes tell you
A positive derivative means the graph rises locally as x increases; a negative one means it falls. Zero derivative means a horizontal tangent, not necessarily a peak or valley: x³ has derivative zero at 0 but keeps increasing through it. A derivative can also fail to exist; the graph of |x| has different left and right slopes at 0. A horizontal tangent does not imply weightlessness.
Try it yourself
For f(x) = x³ + 2x, find f′(x) and its value at x = 2.
Show answer and explanation
Differentiate term by term: f′(x) = 3x² + 2. At x = 2, it is 3 × 4 + 2 = 14. This is a slope, whereas f(2) = 12 is the graph's height.
Use the activity to check your understanding
Move the scanner to a point, predict the sign of the slope and compare the tangent with nearby secant lines.