Estimating a Derivative at a Point

The definition of the derivative needs a formula: you substitute $a + h$, simplify and take a limit. Real data rarely comes with a formula. A thermometer gives readings every few minutes; a graph gives a picture. This page shows how to estimate $f'(a)$ from a table, from a graph and with a calculator, and how to say what the estimate means.

From a table: a difference quotient you can compute

The derivative is the limit of difference quotients as the interval shrinks. With a table, you cannot shrink the interval past the data, so you stop at the best interval the table offers and use that quotient as the estimate.

To estimate $f'(a)$ from a table:

1. Pick two inputs on either side of $a$, as close to $a$ as the table allows: $c < a < b$. If $a$ itself is in the table, the closest inputs just before and just after it are a good choice.
2. Compute $\frac{f(b) - f(c)}{b - c}$, subtracting in the same order on top and bottom.
3. Attach units: output units per input unit.

The interval must contain $a$. A quotient from data far away describes a different stretch of the function. At the first or last entry of a table there is only one side, so use the closest one-sided quotient.

Notice what the answer is: a rate, with "per minute" in its units. A frequent slip is to give a temperature instead, such as averaging $B(10)$ and $B(14)$ to get 113. That estimates $B(12)$, the value, not $B'(12)$, the rate.

Notice, too, how the sentence is built. It names the time ($t = 12$), the quantity (the bread's temperature), the direction (decreasing, from the negative sign) and the units. A sentence that says only "the slope is $-4$" leaves out everything a reader needs.

Uneven spacing is fine

Tables on the exam often have unequal gaps. Use the actual widths. With data at $t = 9$ and $t = 12$, the width is 3, whatever the other gaps are. In Worked example 1, the intervals are 4, 6, 4 and 6 minutes wide; each quotient divides by its own width.

The symmetric difference quotient

If you know $f$ at $a - h$ and $a + h$, the quotient across the point,

$$\frac{f(a + h) - f(a - h)}{2h},$$

is the symmetric difference quotient. Its denominator is the full width $2h$. It equals the average of the left quotient $\frac{f(a) - f(a - h)}{h}$ and the right quotient $\frac{f(a + h) - f(a)}{h}$, and their errors partly cancel, so it is usually much closer to $f'(a)$ than either one.

From a graph

When you have a graph, the derivative at $a$ is the slope of the tangent line at $(a, f(a))$. If the tangent line is drawn, read two points on it where it crosses grid lines and compute rise over run. If it is not drawn, lay a ruler along the curve so that it just touches at the point, then read two points on the ruler. Choose points far apart on the line: small reading errors matter less over a long run.

A graph also tells you the sign of $f'(a)$ at a glance: positive where the graph rises, negative where it falls, zero where its tangent line is level.

With a calculator

A graphing calculator has a numerical derivative command. Give it the function and the input, and it returns a decimal estimate of $f'(a)$, computed from a symmetric difference quotient with a very small $h$. On calculator questions:

• Write the setup, such as $\left.\frac{d}{dx}\left( \frac{e^x}{1 + x^2} \right)\right|_{x=2}$, then the value. A bare number earns little.
• Give three decimal places: $0.296$. Store intermediate values instead of rounding them.
• Use radian mode whenever a trigonometric function appears.

Which estimate to trust

Every one of these is an approximation of a limit. The exam asks for the estimate, its setup and its units; it does not expect the exact value. The next topic asks when a derivative exists at all.