How To Estimate A Derivative From A Graph

Rates of change show up all over in science, and especially in physics through quantities like speed and acceleration. Derivatives describe the rate of change of one quantity with respect to another mathematically, but calculating them can be complicated sometimes, and you might be presented with a graph rather than a function in equation form. If you're presented with a graph of a curve and have to find the derivative from it, you might not be able to be as accurate as with an equation, but you can easily make a solid estimate.

Outside of the abstract setting of differentiating an equation, you might be a little confused about what a derivative really is. In algebra, a derivative of a function is an equation that tells you the value of the "slope" of the function at any point. In other words, it tells you how much one quantity changes given a small change in the other. On a graph, the gradient or slope of the line tells you how much the dependent variable (placed on the ​y​-axis) changes with the independent variable (on the ​x​-axis).

For straight-line graphs, you determine the (constant) rate of change by calculating the slope of the graph. Relationships described by curves aren't as easy to deal with, but the principle that the derivative just means the slope (at that specific point) still holds true.

1. Choose the Right Location for Your Derivative

For relationships described by curves, the derivative takes a different value at every point along the curve. To estimate the derivative of the graph, you need to choose a point to take the derivative at. For example, if you have a graph showing distance traveled against time, on a straight-line graph, the slope would tell you the constant speed. For speeds that change with time, the graph would be a curve, but a straight line that just touches the curve at one point (a line tangential to the curve) represents the rate of change at that specific point.

Choose a spot that you need to know the derivative at. Using the distance traveled vs. time example, select the time at which you want to know the speed of travel. If you need to know the speed at several different points, you can run through this process for each individual point. If you want to know the speed 15 seconds after the start of the motion, choose the spot on the curve at 15 seconds on the ​x​-axis.

2. Draw a Tangent Line to the Curve at That Point

Draw a line tangential to the curve at the point you're interested in. Take your time when doing this, because it's the most important and most challenging part of the process. Your estimate will be better if you draw a more accurate tangent line. Hold a ruler up to the point on the curve and adjust its orientation so the line you draw will ​only​ touch the curve at the single point you're interested in.

Draw your line as long as the graph will allow. Make sure you can easily read two values for both the ​x​ and ​y​ coordinates, one near the start of your line and one near the end. You don't absolutely need to draw a long line (technically any straight line is suitable), but longer lines tend to be easier to measure the slope of.

3. Find the Slope of the Tangent Line

Locate two places on your line and make a note of the ​x​ and ​y​ coordinates for them. For example, imagine your tangent line as two notable spots at ​x​ = 1, ​y​ = 3 and ​x​ = 10, ​y​ = 30, which you can call Point 1 and Point 2. Using the symbols ​x​₁ and ​y​₁ to represent the coordinates of the first point and ​x​₂ and ​y​₂ to represent the coordinates of the second point, the slope ​m​ is given by:

\(m = \frac{y_2 – y_1}{x_2 – x_1}\)

This tells you the derivative of the curve at the point where the line touches the curve. In the example, ​x​₁ = 1, ​x​₂ = 10, ​y​₁ = 3 and ​y​₂ = 30, so:

\(\begin{aligned}
m &= \frac{30 – 3}{10 – 1} \
\,\
&= \frac{27}{9} \
\,\
&=9
\end{aligned}\)

In the example, this result would be the speed at the chosen point. So if the ​x​-axis was measured in seconds and the ​y​-axis was measured in meters, the result would mean that the vehicle in question was travelling at 3 meters per second. Regardless of the specific quantity you're calculating, the process of estimating the derivative is the same.