A first-derivative sign chart is a compact way to show where f′(x) is positive, zero, negative, or undefined. It directly connects the derivative to the increasing and decreasing behavior of f.
Worked example: f(x)=x^4-4x^2
Differentiate:
f′(x) = 4x^3 - 8x
= 4x(x^2 - 2)
= 4x(x - sqrt(2))(x + sqrt(2))
The critical numbers are -sqrt(2), 0, and sqrt(2).
Order the critical numbers
Place them from left to right:
-∞ | -sqrt(2) | 0 | sqrt(2) | +∞
Determine the sign on each interval
| Interval | Sign of f′ | Function behavior |
|---|---|---|
(-∞,-sqrt(2)) |
Negative | Decreasing |
(-sqrt(2),0) |
Positive | Increasing |
(0,sqrt(2)) |
Negative | Decreasing |
(sqrt(2),∞) |
Positive | Increasing |
Read extrema from sign changes
- At
x=-sqrt(2), the sign changes from negative to positive, so there is a local minimum. - At
x=0, the sign changes from positive to negative, so there is a local maximum. - At
x=sqrt(2), the sign changes from negative to positive, so there is another local minimum.
Preserve exact values
Use sqrt(2) rather than immediately replacing it with a decimal. Exact notation keeps the chart mathematically precise and makes symmetry visible.
When a decimal is required, keep the exact value in your working and round only the final reported coordinate or interval boundary.
Undefined derivative values
If f′ is undefined at a point, determine whether the original function is defined there. The point may represent a corner, cusp, vertical tangent, or a discontinuity. Use the appropriate marker instead of automatically writing zero.
Create the chart online
In the Variation Table tool, choose the First Derivative Sign Chart. Enter the function, select the domain or interval, and calculate. Review the critical numbers and signs before exporting.
Verification checklist
- Factor the derivative correctly.
- Include domain boundaries and discontinuities.
- Order every critical number.
- Test one point in each open interval.
- Match each sign to increasing or decreasing behavior.
- Evaluate the original function when extremum values are needed.
Create this derivative sign chart online →
Not sure which table to choose? Compare a variation table and a derivative sign chart.