## Increments

For y = f(x):

Δx = x_{2} - x_{1}sox_{2} = x_{1} + Δx

Δy = y_{2} - y_{1} = f(x_{2}) - f(x_{1}) = f(x_{1} + Δx) - f(x_{1})

- 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*.