A hypothetical cart has position s(t)=t²: t is the number of seconds, s the number of metres, 0≤t≤3. From 2 to 3 seconds its position changes from 4 to 9 metres: average velocity 5 m/s. What is its velocity at 2 seconds? Shrink the interval and take the limit of average velocities: the derivative. Use line slope and limits.

Never divide by zero

Let h be the time increment. For h=0.1 the average velocity is 4.1 m/s; for h=0.01, 4.01 m/s. From the left h=−0.1 gives 3.9 m/s. Both directions approach 4:

(s(2 + h) − s(2))/(h) = ((2 + h)^(2) − 4)/(h) = 4 + h (h ≠ 0).

Cancellation requires h≠0. Zero describes where h approaches, not a permitted denominator. The general definition is:

f′(a) = (lim⁡)_(h→0)(f(a + h) − f(a))/(h).

f′(a) is the derivative of f at a. If this finite two-sided limit exists, f is differentiable there. These values form the derivative function f′. Our model gives s′(2)=4 m/s; derivative units are output units divided by input units. OpenStax

Secants approach a tangent

s=t²; — m=4; -- m=5; ·· m=4.5

0123049t, ss, m
Hypothetical s=t², seconds and metres. Orange parabola; solid tangent slope 4, dashed secant through t=3 slope 5, dotted through t=2.5 slope 4.5. Uniform scales; window 0≤s≤10.

A secant goes through two graph points and its slope measures average change. As the second point approaches the first, slopes 5 and 4.5 approach 4. The line with that limiting slope is the tangent. It gives the local direction; it need not be a line meeting the whole graph at only one point.

Equation of a tangent

At (a,f(a)), use that point and slope f′(a):

y = f(a) + f′(a)(x − a); s_(tan)(t) = 4 + 4(t − 2) = 4t − 4.

At t=2, the tangent has the same value 4 and slope 4. Far away it cannot replace the parabola. The instantaneous velocity is exact within our hypothetical formula; estimates from measured data retain measurement errors even over short intervals.

When the definition fails

For f(x)=|x| at zero the quotient is −1 from the left and 1 from the right. The function is continuous but has no single derivative. Differentiability implies continuity; the converse is false. Connection and exceptions.

Thus 4 m/s at 2 seconds follows from change over successively shorter intervals. Check the model domain, units and existence of a common finite limit before using a derivative.