# Differentiable Functions and Their Derivatives

• G. Baley Price

## Abstract

One of the important problems in mathematics and in its applications in science and engineering is the following: if y = f(x), find the rate of increase of y with respect to x. For example, if y = 2x + 5, then y0 = 2x0 + 5, y1 = 2x1 + 5, and y1y0 = 2(x1>x0). Thus
$$\frac{{{y_1} - {y_0}}}{{{x_1} - {x_0}}} = 2,$$
(1)
(1) and the rate of increase of y with respect to x is 2. This example shows that the problem has a simple solution in all cases in which f is a linear function. Thus, if y = ax + b, then
$$\frac{{{y_1} - {y_0}}}{{{x_1} - {x_0}}} = 2,$$
(2)
Observe that, for this linear function f, the rate of increase of y with respect to x is the same, namelya, for everyx0 and xl.

## Keywords

Partial Derivative Linear Transformation Differentiable Function Tangent Plane Chain Rule
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.