Differentials

Small changes, estimated

You measure a ball's radius as 10 cm, but your ruler could be off by half a millimetre. How far off could your calculated volume be?

When x changes by a small amount dx, the true change in y is \Delta y, measured along the curve. The differential dy is the change along the tangent line instead, and it is much easier to compute. Shrink dx and compare the ochre and indigo segments.

Drag the ochre point to change dx. The ochre segment is Δy along the curve. The indigo segment is dy along the tangent.

dy = f′(x) dx
dx
Actual change Δy
Differential dy
Δy − dy

The differential

For y=f(x), the differential is

dy=f'(x)\,dx.

Here dx is any change in x, and dy is the matching change along the tangent line. For small dx, \Delta y\approx dy. This is linear approximation written in terms of changes rather than values, and it explains the notation \frac{dy}{dx}: the derivative is the ratio of the two differentials.

Estimating error

If a measured quantity x has an error of at most dx, then a quantity computed from it, y=f(x), has an error of about dy. For the ball, V=\tfrac43\pi r^3 gives

dV=4\pi r^2\,dr=4\pi(10)^2(0.05)=20\pi\approx 62.8\ \text{cm}^3.

Comparing with the volume itself gives the relative error, \frac{dV}{V}=\frac{3\,dr}{r}=0.015, or 1.5%. Notice that a 0.5% error in the radius became a 1.5% error in the volume. Where did the factor of 3 come from?

Solve it step by step

Compute each differential, and use it to estimate a change or an error. Answers can use pi, such as 20pi. The score counts steps you answer correctly on the first try.

Problems solved0
First-try steps0 of 0

    Common mistakes

    Confusing Δy and dy

    \Delta y is the exact change along the curve. dy is the estimate along the tangent. They agree only approximately.

    Evaluating the derivative at x + dx

    Use the slope at the starting point x. That is where the tangent line touches the curve.

    Mixing absolute and relative error

    dV is in cubic centimetres. \frac{dV}{V} has no units, and is often given as a percentage.