Differentials are a compact way to describe how a small change in an input produces an approximate change in an output. In calculus, dx represents a small chosen change in x, while dy represents the change predicted by the tangent line to y = f(x). This idea matters because many real measurements and calculations involve tiny changes, not perfectly exact values.
Differentials turn the derivative into a practical tool for estimation.
At a point x = a, the derivative f'(a) gives the slope of the tangent line, so the differential satisfies dy = f'(a) dx. The actual change in the function is Δy = f(a + dx) - f(a), which is usually close to dy when dx is small and the curve is smooth. This leads to the linear approximation f(a + dx) ≈ f(a) + f'(a) dx.
Differentials are also used to estimate propagated error, such as how uncertainty in a radius affects the uncertainty in area or volume.
Key Facts
- dx is a small change in the input x, often chosen by the user or measurement situation.
- dy is the tangent-line change in y, given by dy = f'(x) dx.
- The actual change is Δy = f(x + dx) - f(x), which is not always equal to dy.
- Linear approximation: f(a + Δx) ≈ f(a) + f'(a) Δx.
- For small changes, Δy ≈ dy when f is differentiable and Δx is close to 0.
- Estimated error in a function value can be written as |dy| = |f'(x)| |dx|.
Vocabulary
- Differential
- A differential is an expression that estimates a small change in a quantity using a derivative.
- dx
- dx is a small change in the independent variable x.
- dy
- dy is the approximate change in y predicted by the tangent line, calculated as dy = f'(x) dx.
- Tangent line
- A tangent line is the line that touches a curve at a point and has slope equal to the derivative there.
- Linear approximation
- Linear approximation estimates a function near a point by using the value and slope of the tangent line.
Common Mistakes to Avoid
- Treating dy and Δy as always equal is wrong because dy is the tangent-line estimate while Δy is the actual change on the curve.
- Forgetting to evaluate the derivative at the starting point is wrong because dy = f'(a) dx uses the slope at x = a, not at a random nearby value.
- Using a large dx without checking accuracy is wrong because linear approximation becomes less reliable as the input change gets farther from the base point.
- Ignoring units in differentials is wrong because dx and dy carry the units of their variables, and the derivative carries output units per input unit.
Practice Questions
- 1 For f(x) = x^2 at x = 3, use differentials to estimate the change in y when dx = 0.1. Then compare with the actual change Δy.
- 2 The radius of a circle is measured as r = 10 cm with a possible error of dr = 0.05 cm. Use dA = 2πr dr to estimate the possible error in the area.
- 3 A curve is concave up near x = a and dx is positive. Explain whether the tangent-line differential dy is likely to overestimate or underestimate the actual change Δy.