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The Differential Calculator

Differential calculator. Enter f(x), x and dx to compute dy=f′(x)dx, the true increment Δf and the approximation error—for local linearization.

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dy = f′(x)·dx, f(x+dx) ≈ f(x) + dy
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Base point
Smaller is better

📖 Tutorial｜The Differential

1. Definition
The differential dy=f′(x)dx is the linear part of the increment Δf; for small dx, f(x+dx)≈f(x)+dy.
2. Symbols
SymbolMeaning
dxIncrement of x
dyDifferential (linear increment)
ΔfTrue increment
f′(x)Tangent slope
3. How it works
  • We numerically compute f and f′;
  • Compute dy=f′dx;
  • Compare with the true Δf and error.
4. Steps
  1. Enter f(x);
  2. Enter base point x and increment dx;
  3. Click Calculate.
5. Example
Example: f=x² at x=1, dx=0.1. f′=2, dy=0.2, f(1.1)≈1.2, true 1.21, error 0.01.
6. Pitfalls
Smaller dx gives a better linear approximation;
dy is the tangent increment, Δf the curve increment;
Differentials are used for approximations and error estimates.

❓ FAQ｜The Differential

Relation between differential and derivative?
The derivative is f′=dy/dx; the differential is dy=f′dx.
Why is the linear approximation accurate only for small dx?
Near the tangency the curve looks like its tangent; large dx ignores curvature.
dy vs Δf?
dy is the tangent increment; Δf is the true curve increment; their difference is a higher-order term.
Practical uses of differentials?
Approximating values, estimating measurement errors, deriving approximation formulas.
Why is f(1.1) not 1.2?
The linear estimate is 1.2 but the true value is 1.21; the 0.01 gap is the second-order term.