Differential of a function

For other uses of differential in calculus, see differential (calculus), and for more general meanings, see differential.

In the field of mathematics called calculus, the differential applys to a function and is defined as the result of a product of two values, one of wich is obtained throught the derivation of the function.

If y represents a function, then its differential is usually noted:

$\mathrm {d} y\,$

Not to be confused with Leibniz's notation.

In a few words, the differential of a function at a point is the product of the derivative of that function at that point by another arbitrary value.

Definition

The definition of a differential is given by this equality:

${\mathrm {d} f(x)=f'(x)\Delta x}\,$

In words "the diferential of a function at a point is equal to the result of the product between, the value of the derivative of the function at the point, and the value of $\Delta x$"

From the equality we can see that $\mathrm {d} f(x)$ is a dependant variable. In the other side, $x$ and $\Delta x$, both are independent variables. That is, the differential is a function of two independent variables.

The variable $\Delta x$ is interpreted as an increment of the variable $x$.

The differential of the identity function

There is an interesting relation that appears from applying the differential to the identity function $f(x)=x$:

If ${f(x)=x}$ then:

${\mathrm {d} f(x)=\mathrm {d} x=f'(x).\Delta x}\,$

but ${f'(x)=\mathrm {D} _{x}[x]=1}\,$, then

${\mathrm {d} f(x)=\mathrm {d} x=1.\Delta x}\,$

and so

${\mathrm {d} x=\Delta x}\,$

The geometric meaning of the differential

Geometric interpretation of the differential of a function $f(x)$ at a point $x_{0}$.

The differential can be taken in a geometric sense as the amount that the tangent line rises or falls from the point in wich the differential is taken.

Remember that the derivative of a function at a point is the slope of the tangent line for the function at the point, multiplying this slope by a number we get the rise of the tangent line.

In the geometric sense, the rise is vertically from the point in wich the differential is taken. The increment $\Delta x$ taken represents the horizontal remoteness from the point.

This way the elevation of the tangent line obtained as a result will depend on the actual point and horizontal remoteness taken, wich in the mathematical formulas are symbolized $x$ and $\Delta x$ respectivelly.