Calculus I · 2B · lesson
Differentials
Learn differentials with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Local linearity, differentials, and Newton's methodWhat this section is building
Learn differentials with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.
Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.
Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.
Learning objectives
Use to estimate changes in an output.
Differentials and Small Changes
Before the formulas
The approximation in Differentials replaces a nonlinear function with its best local line. The known value supplies the starting height, and the derivative supplies the rate of change. The estimate is useful because lines are easy to calculate with, not because the original curve has become exactly linear.
Always identify the center and the target . Write , calculate the predicted change , and then add it to . Use concavity or a numerical comparison to understand the direction and size of the error.
Read this graph as text
Differentials separate the actual change from the tangent-line estimate. The curve produces the actual output change y . The tangent line produces the differential dy=f'(x)dx . Their vertical gap is the approximation error. Starting at x=1 , the input changes by dx=0.7 . The tangent predicts dy=2(0.7)=1.4 , while the curve actually changes by y=1.89 . The two quantities are close only when the input change is sufficiently small; they are not identical by definition.
Every relationship in differentials separate the actual change from the tangent-line estimate is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: This visual prevents the common equation dy= y from being treated as exact. It should reinforce that dy belongs to the tangent model and y belongs to the original function.
The curve produces the actual output change y. The tangent line produces the differential dy=f'(x)dx. Their vertical gap is the approximation error.
Differentials package the tangent-line prediction
For an input change , the differential is the change predicted by the tangent line. The actual change is . Near the base point, and are close but not generally equal.
This notation is especially useful in measurement problems because it keeps the estimated output error attached to the input uncertainty and local sensitivity.
Differentials package the linear approximation into the compact relation
Here represents a chosen small input change, and represents the corresponding linearized output change. The actual change is usually close to, but not exactly equal to, .
This notation is especially useful when several measurements contribute to an engineering calculation. It preserves units and makes first-order error propagation visible.
If , define the input differential as a chosen small change in . The corresponding differential in is
The actual output change is
For small ,
Estimate a change in area
A circular plate has radius cm. Estimate the change in area if the radius increases by cm.
Worked solution
Write a real attempt before opening the supplied answer.
Derivative as a local conversion factor
The relation says that near a chosen input, the derivative converts a small input change into an approximate output change.
Volume added by a thin coating
A metal sphere of radius cm receives a coating of thickness cm. Since
we estimate
This is the surface area times thickness, exactly the geometric shell interpretation expected for a very thin coating.
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