Meaning of the Differential

Meaning of the Differential

Differential of a Function of One Independent Variable

Note: Assume all functions in the lesson are differentiable on an interval.

Let $y = f(x)$ define $y$ as a function of $x$. Then its derivative $f’(x)$ will give the slope of the curve at any point $P(x,y)$ on it, i.e., it is the slope of the tangent line drawn to the curve at $P$.

From figure 6.12, it’s evident that:

\[f'(x) = tan\alpha = \frac{dy}{\Delta x} \tag{6.1}\]

Hence,

\[dy = f'(x)\Delta x \tag{6.11}\]

We call $dy$ the differential of $y$, i.e. it is the differential of the function defined by $y = f(x)$.

Note that the differential of $y$ ($dy$) is dependent on the abscissa $x$ and on the size of $\Delta x$. That means while $y = f(x)$ defines $y$ as a function of one independent variable $x$, the differential $dy$ is a function of two independent variables - $x$ and $\Delta x$.

Definition 6.13 The differential of y Let $y=f(x)$ define y as a function of x on an interval $I$. The differential of y, writen as $dy$ (or $df$) is defined by

\[(dy)(x,\Delta x) = f'(x)\Delta x \tag{6.14}\]

Differential of a function of Two Independent Variables

Definition 6.4 Differential of z Let $z = f(x,y)$ define $z$ as a function of $x$ and $y$. The differential of z, written as $dz$ or $df$, is defined by:

\[(dz)(x,y,\Delta x,\Delta y) = \frac{\partial f(x,y)}{\partial x}\Delta x + \frac{\partial f(x,y)}{\partial y}\Delta y \tag{6.41}\]

Differential Equations with Separable Variables

The first order differential equations in this chapter can be written in the form:

\[Q(x,y)\frac{dy}{dx} + P(x,y) = 0 \tag{6.6}\]

Written in this form it is assumed that $y$ is the dependent variable and $x$ is the independent variable. If we multiply 6.6 by $dx$, it becomes

\[P(x,y)dx + Q(x,y)dy = 0. \tag{6.61}\]

Written in this form, either $x$ or $y$ may be considered as being the dependent variable. However, in both cases, $dy$ and $dx$ are differentials and not incremements.

These two forms don’t cover all equations of the first order, but they are inclusive enough to cover most of the applications we will meet.

Separable Variables

If it is possible to rewrite (6.6) or (6.61) in the form

\[f(x)dx + g(y)dy = 0 \tag{6.62}\]

so that the coefficient of $dx$ is a function of $x$ alone and the coefficient of $dy$ is a function of $y$ alone, the variables are called separable. And after they have been put in the form (6.62), they are said to be separated. A 1-parameter family of solutions of (6.62) is then

\[\int f(x)dx + \int g(y)dy = C, \tag{6.63}\]

where $C$ is an arbitrary constant.