The notation of dy/dx has been regarded as a single symbol for a derivative in previous sections. It looks like a fraction but is not one. In this section, dy and dx, named differential, are given a meaning so that their quotient is the derivative of the function.

Let y = f(x) and is differentiable. If the differential dx is an independent variable, then the differential dy is defined in terms of dx as dy = f ^'(x)dx. Thus

     dy/dx = f ^'(x)

Let A(x, f(x)) and D((x + Δx), f(x + Δx)) be points on function y. Let dx = Δx. Then the change Δy equals f(x + Δx) - f(x).

The slope of the tangent line AB is the derivative of function y, that is

In this equation dy is the amount of changes in y axis along the tangent line to the curve at point A, and dx is the changes in x axis which is Δx.

The definition of derivative gives

therefore,

that is

when Δx approaches 0. In this equation dx = Δx, so

therefore,

     dy ≈ Δy.

Differential can calculate the approximate value of another poiont nearby. In the diagram, the coordinate of point B is (a + Δx, f(a + Δx)). The value of its y coordinate is

     f(a + Δx) = f(a) + Δy

When Δx is small dy ≈ Δy and f(a + Δx) ≈ f(a) + dy . This formula is commonly used to calculate approximate values of functions.

This concept can be understood by calculating the the value of Δy and dy when the value of x changes from 2.02 to 2.03 for the function of y = x³ - 2x² + 4.

Substitute a = 2.02 into y = y = x³ - 2x² + 4,

     f(a) = f(2.02) = 2.02³ - 2(2.02)² + 4 = 4.081608

so point A(2.02, 4.081608)

     f(a) = f(2.03) = 2.03³ - 2(2.03)² + 4 = 4.123627

This gives point B(2.03, 4.123627)

     Δy = f(2.03) - f(2.02)

          = 4.123627 - 4.081608

          = 0.042019

The derivative of y is

     y ^' = (x³ - 2x² + 4) ^' = 3x² - 6

According to differential definition,

     dy = y ^'dx = (3x² - 6)dx

In this example, dx = Δx = 2.03 -2.02 = 0.01. Substituting dx = 0.01 and x = 2.02 into equation dy gives

     dy = (3x² - 6)dx = (3(2.02)²-6)(0.01) = 0.062412

Notice that Δy and dy are very close in this case.The formula

     f(a + Δx) ≈ f(a) + dy

or

     f(a + Δx) ≈ f(a) + f ^'(x-a)

is called the linear approximation of the function at point a.