]> Sign of Derivative

Sign of Derivative

and the tangent line of a function.

by Samuel Dagan (Copyright © 2007)


From the definition, if at a given point of the independent variable  x  the derivative is positive(negative), the function must be strictly increasing(decreasing) there. In this case, the attribute "strictly" will be omitted.

By using the mathematical language the positive(negative) sign of a derivative at a given point is sufficient (but not necessary) condition for a function to be increasing(decreasing) there.

If the derivative is zero at a given point, it means that the tangent line is parallel to the x axis and locally the function does not change. Such point is called a stationary point of the function, and is defined as a root of the derivative.

Let's take a closer look at a srictly increasing(decreasing) function in an interval, but with one stationary point. This means that the derivative is positive(negative) for all the interval, except for this point. Therefore the derivative itself has a minimum(maximum) at that particular point.

Let's take the following example:

y= x 3 dy dx =3 x 2 } (1)

This function is strictly increasing and its derivative is positive except at point  x = 0 , where the derivative has a minimum.

The graphic presentation of this example (1) is at Fig. Derivative of x³ .

A stationary point is obtained at a (local) maximum(minimum) of a differentiable function, since the derivative is positive(negative) at the left-hand side of the point, and - is negative(positive) at the right-hand side. In other words the derivative should intersect the x axis at that point. By using mathematical language a stationary point is a necessary, but not sufficient condition for a maximum(minimum) of a differentiable fuction.

We have learned so far how to look at the sign of a derivative in order to study if a function is increasing or decreasing, but we cannot specify what happens at a stationary point without knowing how the derivative is changing there. In order to learn if a derivative is increasing or decreasing, we need to know the derivative of the derivative, which is called the second order derivative and commonly - the second derivative.