Differentials

The differential of a function

• For y = f(x) the differential is dy = f′(x) dx: the differential of the function equals its derivative multiplied by the differential of the independent variable.
• dx is the increment of x; dy is the corresponding change along the tangent, not along the curve. For small increments dy ≈ Δy, and the error shrinks as Δx → 0.
• Second differential: d²y = f″(x) dx² (only when x is the independent variable, so that dx is constant).
• Rewriting a differential equation in differential form is what makes direct integration legal: dy/dx = g(x)h(y) → dy/h(y) = g(x) dx, then integrate both sides.

Worked examples

1. y = x³ ln x → dy = (3x² ln x + x²) dx.
2. y = sin(2x + 1) → dy = 2 cos(2x + 1) dx.
3. Approximation: a cube of side 5 cm grows by 0.1 cm. V = x³, dV = 3x² dx = 3(25)(0.1) = 7.5 cm³ (the true increase is 7.31 cm³).
4. dy/dx = x/y with y(0) = 3 → y dy = x dx → y²/2 = x²/2 + C → C = 9/2, so y² = x² + 9.