Difficulty: Introductory | Prerequisites: Limit definition of the derivative (Section 2.1 core material), reading graphs.
Once you can compute a derivative exactly from the limit definition, the next questions are: what do you do when you only have a table of values instead of a formula, and does the derivative always exist? This material rounds out Section 2.1 by teaching you to approximate derivatives from data and to recognise situations where the derivative fails to exist, particularly at corners where the graph has a sharp change in direction. Both ideas recur throughout the course and appear regularly on exams.
When you have data points instead of a formula, you approximate the derivative by computing difference quotients for small values of h and watching what they approach. A derivative exists at a point only if the left-hand and right-hand slopes agree; at a sharp corner or cusp they do not, and the derivative is undefined there.
Numerical approximation of the derivative
Estimating f'(a) by computing [f(a+h) – f(a)] / h for one or more small values of h, using a table or calculator rather than an algebraic formula. Think of it as getting close to the exact slope by using nearby data points.
Forward difference quotient
The approximation [f(a+h) – f(a)] / h with h > 0. It uses a point to the right of a.
Backward difference quotient
The approximation [f(a+h) – f(a)] / h with h < 0. It uses a point to the left of a. Both forward and backward quotients should converge to the same value if the derivative exists.
Velocity
The derivative of a position function with respect to time: v(t) = f'(t). Units are the position units divided by the time units (e.g. feet per second). In simple terms, velocity is how fast and in which direction something is moving at an instant.
Non-differentiability (derivative does not exist)
A function is not differentiable at a point where the limit defining the derivative fails to exist. Common causes include corners (sharp bends), cusps, vertical tangent lines, and discontinuities. In simple terms, if the graph has a sharp point or a break, you cannot draw a single tangent line there.
Corner (sharp turn)
A point where the graph changes direction abruptly, so the slope approaching from the left is different from the slope approaching from the right. The left and right limits of the difference quotient both exist but disagree, so the derivative does not exist.
When you have a table of (x, f(x)) values but no formula, you estimate f'(a) by computing [f(a+h) – f(a)] / h for the smallest available h values.
Worked example: f(x) = 5x³, values near x = 2.
h | f(2+h) | [f(2+h) – f(2)] / h |
|---|---|---|
0.1 | 46.305 | (46.305 – 40) / 0.1 = 63.05 |
0.01 | 40.603 | (40.603 – 40) / 0.01 = 60.3 |
0.001 | 40.06 | (40.06 – 40) / 0.001 = 60 |
–0.01 | 39.403 | (39.403 – 40) / (–0.01) = 59.7 |
The exact derivative is f'(2) = 60. As h gets smaller, the approximations close in on 60 from both sides.
To decide which h values are "small enough," compare each approximation to the exact value. The problem asks for accuracy to 1 decimal place (error ±0.05). Here h = 0.001 gives exactly 60, and h = –0.01 gives 59.7, both within 0.05 of 60. The values h = 0.01 and h = 0.001 and h = –0.01 all qualify.
When f(t) gives position at time t, the derivative f'(t) gives velocity at time t.
Units: velocity = (position units) / (time units). If position is in feet and time in seconds, velocity is in feet per second.
Sign: a negative derivative means the function is decreasing. For a falling object, f'(t) < 0 means the height is going down, which makes physical sense.
Example: A tomato dropped from y = 10 feet at t = 0, landing at t = 0.8.
For 0 < t < 0.8, velocity is negative, so the height is decreasing (the tomato is falling).
For t > 0.8, velocity is zero, because the tomato is on the ground and not moving.
The derivative f'(a) exists only when the limit lim(h→0) [f(a+h) – f(a)] / h exists. That limit fails to exist when the left-hand limit (h → 0⁻) and the right-hand limit (h → 0⁺) give different values.
At the tomato's landing point t = 0.8:
Left slope (just before landing, t in [0.75, 0.8]): the graph is steep and falling. Estimating rise/run from the graph gives a slope around –20 (the exact number depends on your graph reading).
Right slope (just after landing, t in [0.8, 0.9]): the tomato is on the ground, height = 0, so the slope is 0.
The two slopes are different (–20 from the left, 0 from the right), so f'(0.8) does not exist.
This is a corner: the graph makes a sharp turn at the landing point. Neither the left slope alone nor the right slope alone is "the" derivative. The derivative simply does not exist at a corner.
f'(a) \approx \frac{f(a+h) - f(a)}{h} \quad \text{for small } hNumerical approximation: use small positive h (forward) or small negative h (backward). Smaller h generally gives a better approximation.
\text{Error} = \left| \frac{f(a+h) - f(a)}{h} - f'(a) \right|The approximation is "accurate to 1 decimal place" when this error is at most 0.05.
In practice, many quantities (temperature readings, stock prices, sensor data) come as tables, not formulas. Engineers and scientists estimate rates of change from sampled data using exactly this difference-quotient approach. The question of when a derivative fails to exist also matters physically: at the moment an object hits the ground, a sharp transition occurs and the velocity is undefined at that instant.
Students often think a smaller h always gives a dramatically better answer. In practice, rounding in table values can make very small h less accurate (this matters more in applied work, but be aware).
Thinking the derivative must exist everywhere a function is defined. Continuity does not guarantee differentiability. The tomato's height function is continuous at t = 0.8 but not differentiable there.
Believing that if either the left-hand or right-hand slope exists, the derivative exists. Both must exist and be equal.
Confusing "the velocity is zero" with "the derivative does not exist." After the tomato lands, velocity is zero (the derivative exists and equals zero). At the landing instant, the derivative does not exist because the two one-sided slopes disagree.
⚠️ Table-based approximation problems test whether you can apply the difference quotient formula to data, not just to formulas. Expect at least one on a midterm.
⚠️ "Does the derivative exist?" questions are common on exams. The answer almost always hinges on whether the left and right slopes match.
⚠️ Know your units. If position is in feet and time in seconds, the derivative (velocity) is in feet per second. Unit questions are easy marks if you know the rule: derivative units = (numerator function units) / (denominator variable units).
⚠️ Negative velocity means the function is decreasing. Do not confuse negative velocity with "slowing down" (that involves acceleration, a later topic).
True or False: If f'(a) = –3, the function f is decreasing at x = a. (True.)
Fill in the blank: The units of the derivative of a position function measured in metres over time in seconds are ___. (Metres per second.)
True or False: A function can be continuous at a point but not differentiable there. (True, e.g. a corner.)
Fill in the blank: For the derivative to exist at a point, the left-hand and right-hand ___ must be equal. (Slopes / limits of the difference quotient.)
True or False: Using h = 0.001 always gives a better approximation than h = 0.01. (Generally true for well-behaved functions, but not guaranteed when table values are rounded.)
Q: A table gives f(3) = 12, f(3.01) = 12.07, f(2.99) = 11.94. Estimate f'(3) using h = 0.01 and h = –0.01.
A: Forward: (12.07 – 12) / 0.01 = 7. Backward: (11.94 – 12) / (–0.01) = 6. A reasonable estimate is around 6.5, and you would need more data or the exact formula to pin it down.
Q: A ball is thrown upward. Its height in metres at time t seconds is given by a graph. At t = 2, the ball reaches its peak. What is f'(2)?
A: f'(2) = 0. At the peak, the tangent line is horizontal.
Q: A piecewise function is defined as f(x) = x² for x < 1 and f(x) = 2x – 1 for x ≥ 1. Is f differentiable at x = 1?
A: Left slope at x = 1: the derivative of x² is 2x, which gives 2. Right slope: the derivative of 2x – 1 is 2. Since both sides give 2 and f is continuous at 1, f is differentiable there with f'(1) = 2.
Q: A function has a sharp corner at x = 4 where the slope from the left is 3 and the slope from the right is –1. Does f'(4) exist?
A: No. The left and right limits of the difference quotient are 3 and –1, which are not equal, so the derivative does not exist at x = 4.
Numerical approximation reappears in Section 3.9 (linear approximation and differentials), where the tangent line itself becomes a tool for estimating function values. Non-differentiability connects to Section 2.5 (differentiability implies continuity, but not the reverse) and to later work on the Mean Value Theorem, which requires differentiability on an interval.
numerical derivative approximation, forward difference, backward difference, difference quotient table, derivative does not exist, non-differentiable, corner point, cusp, left-hand slope, right-hand slope, velocity as derivative, feet per second, derivative units, piecewise function differentiability, MTH 132 Section 2.1, Calculus I Michigan State University