Differentials

The central idea

For a differentiable function $y=f(x)$, the differential is $dy=f'(x)dx$. Here $dx$ is a chosen input change, while $dy$ is the linearized output change. For an actual finite change $\Delta x$, the exact output change is $\Delta y=f(x+\Delta x)-f(x)$, and $\Delta y\approx dy$ when $\Delta x$ is small.

Definitions, hypotheses, and notation

The notation $dy=f'(x)dx$ is an equality between differentials, whereas $\Delta y\approx dy$ is an approximation between an exact finite change and its linear part. Confusing those statements hides the error term. For a linear function there is no curvature, so the approximation becomes exact for every $dx$; for nonlinear functions it improves as the chosen change shrinks.

In error propagation, if a measured input has possible error magnitude $|dx|$, then $|dy|=|f'(x)||dx|$ estimates the resulting absolute output error. Dividing by $|f(x)|$ gives an estimated relative error. This use requires distinguishing a signed change from a maximum error magnitude and preserving the measurement units through multiplication.

Conceptual meaning

A differential is the change predicted by the tangent line. Differentiability means the error is small relative to $dx$: $f(x+dx)=f(x)+f'(x)dx+o(dx)$. Thus differentials support approximation and first-order propagation of measurement error.

A dependable method and decision rule

1. Choose a convenient base point $x$ near the target value.
2. Compute $f(x)$ and $f'(x)$ at that base point.
3. Set $dx$ equal to target input minus base input.
4. Calculate $dy=f'(x)dx$ and use $f(x+dx)\approx f(x)+dy$.
5. Report whether $dy$ is signed change, magnitude, or an error bound estimate.

Fully worked example

Graphical or geometric meaning

The tangent line at $(4,2)$ rises by $dy=0.025$ when its horizontal run is $dx=0.1$. Because $\sqrt x$ is concave down, its tangent line lies above the graph, predicting an overestimate.

Common mistakes and why they fail

Verification and reasonableness checks

• Compare the approximation with a calculator only after deriving it.
• Use concavity to predict whether the tangent approximation is high or low.
• Check units: $f'(x)dx$ must have the same units as $y$.

What the approximation symbol promises

The relation $dy=f'(x)\,dx$ leads to $\Delta y\approx dy$ for a small change $\Delta x=dx$, but not to an exact identity in general. The error is the gap between a curve and its tangent; for a twice differentiable function it is typically proportional to $(\Delta x)^2$ near the base point. Compute $f(x)$ and $f'(x)$ at that base, then use $f(x+\Delta x)\approx f(x)+f'(x)\Delta x$. Units remain consistent because derivative units multiplied by input units give output units. Large curvature or a large step warns that the linear estimate may be poor, so round the result to realistic precision.

Practice

1. For $y=x^2$ at $x=3$ with $dx=0.02$, find $dy$.
2. Use differentials to approximate $1/9.1$ from $x=9$.
3. Is $dy=\Delta y$ exactly for a linear function?

Connections and next steps

• Linear Approximation
• Derivative as a Rate of Change
• Related Rates