Estimate dy from a small dx and the derivative at a base point. The displayed differential estimate includes enough working to inspect signs and scale.
Differentials support measurement propagation, quick sensitivity checks, and local error estimates.
The differential dy=f′(a)dx approximates the actual function change for a small input increment.
On this page, the displayed differential estimate follows this definition rather than any similarly named measure.
For x³ at a=2 with dx=.01, dy≈12(.01)=.12 and the estimated new value is 8.12.
The approximation omits quadratic and higher terms, so large dx or strong curvature increases error. If the Differential Approximation assumptions do not fit, consider target estimate.
When differential approximation looks surprising, restore the sample and vary base point a by itself.
Estimate the derivative, multiply by dx, and add dy to f(a) when a new value is needed.
Write each source value under its matching label before calculating: Function f(x), Base point a and Input change dx. This preserves the assumptions behind differential estimate and makes a later check possible without reopening the original problem.
Reversing dx reverses the first-order change, though actual nonlinear errors need not be symmetric. This relationship remains useful even when the final differential estimate is rounded.
Linear approximation expresses the same idea as a target-value formula. The two results may share inputs while retaining different meanings.
Before publishing or sharing differential estimate, decide on units, significant digits, and whether an exact form is expected. Those choices belong to interpretation rather than the calculator engine.
When reporting the answer, state the differential estimate first, then its value and unit. Add Function f(x), Base point a and Input change dx if someone else must verify the work independently.
In Differential Approximation, the entered fields are Function f(x), Base point a, and Input change dx. Distinguish numerical evidence from symbolic proof. The method can provide a strong practical estimate for a well-behaved function, but it does not establish global identities, exclude every hidden discontinuity, or certify convergence by itself.
For this differential approximation result, before archiving the answer, label its role in the problem and retain the inputs needed to regenerate it. That creates a clearer audit trail than saving an isolated decimal.
Usually not.
Yes.
Higher derivatives and step size.