Section 2.6: Differentials

Increments

For y = f(x):

Δx = x₂ - x₁sox₂ = x₁ + Δx

Δy = y₂ - y₁ = f(x₂) - f(x₁) = f(x₁ + Δx) - f(x₁)

• Δy represents the change in y corresponding to a change Δx in x.
• Δx can be either positive or negative.

Differentials

If y = f(x) defines a differentiable function, then the differential dy, or df, is defined as the product of f'(x) and dx, where dx = Δx. Symbolically,

dy = f'(x) dxordf = f'(x) dxwheredx = Δ x

Approximations Using Differentials

For small Δx,

Δy/Δx ≈ f'(x) and Δy ≈ f'(x)Δx.

Since dy = f'(x) dx, it follows that:

Δy ≈ dy

So, dy can be used to approximate Δy.