The second derivative is the tool that allows us to understand not only the rate of change of the function, but also the rate of change of the rate of change itself. If the first derivative describes the slope of the function at any point, the second derivative describes how the gradient changes—that is, it provides information about the “curvature” or convexity of the function.
The second derivative is denoted as \(f”(x)\), and it helps us understand the behavior of the function in different regions of it, identify types of extremes, and understand whether the function is facing up or down at different points.
A derivative is a function like any function and also from it which derives a derivative. When deriving a derivative from a derivative, the first is called the first derivative and the last is called the second derivative. The chain of functions and their symbolism is as follows:
- Original function: ‘f(x)'
- First derivative: ‘f'(x)'
- Second derivative: ‘f”(x)'
The Meaning of the Second Derivative
The second derivative gives expression to the trend of changes in the slope of the source function. When the second derivative is positive, the slopes of the source function are increasing. The rate of growth in the slopes is in accordance with the result obtained. If the result is 2, then the slope increases by 2 with each additional step.
Main uses of the second derivative
1. Identify the type of extremes
When the first derivative is reset at a certain point, i.e. \( f'(x) = 0 \), it may be an extreme point – maximum or minimum. To know if this is a maximum point or a minimum, we will use the second derivative:
-
If \( f”(x) > 0 \) at the point where \( f'(x) = 0 \):
The function is concave upwards at that point, meaning we have a local minimum point. -
If \( f”(x) < 0 \) at the point where \( f'(x) = 0 \):
The function is concave downward at that point, meaning we have a local maximum point. -
If \( f”(x) = 0 \) at the point where \( f'(x) = 0 \):
The second derivative does not provide enough information, and further examination is needed.
Graph with extremes for a quadratic function that shows a maximum at a particular point
2. Determining the curvature of the function
The second derivative can help us understand the curvature of the function – whether it is “upwards” or “downwards”:
- If \( f”(x) > 0\) is in a certain domain, the function is concave upwards in that domain (like a smile).
- If \( f”(x) < 0\) in a certain domain, the function is concave downwards in that domain (such as a sad face).
Graph of a simple quadratic function showing upward curvature throughout the range
3. Identifying torsion points
A twisting point is a point at which the function changes its concaveness – going from being concave upwards to being concave downwards, or vice versa. At such points, the second derivative is equal to zero, and there is a change in the sign of the second derivative around the point.
Dogma:
If we look at the function \( f(x) = x^3 \), we can see that the function changes its convexity at the point \( x = 0 \), and therefore it is an inflection point.
Example: Calculating a second derivative to find an extreme type
Suppose we have the function \( f(x) = x^3 – 3x \), and we want to find its extremes and understand the type of extremes.
We find the first derivative: \( f'(x) = 3x^2 – 3 \)
Let's look for points where \( f'(x) = 0 \):
\( 3x^2 – 3 = 0 \)
\( x^2 = 1 \)
\( x = \pm 1 \)
That is, we may have extremes in \( x = 1 \) and \( x = -1 \).
Calculate the second derivative to check the type of extreme: \( f”(x) = 6x \)
Let's check each point:
– For \( x = 1 \):
\( f”(1) = 6 \cdot 1 = 6 > 0 \)
That is, at \( x = 1 \) there is a local minimum point.
– For \( x = -1 \):
\( f”(-1) = 6 \cdot (-1) = -6 < 0 \)
That is, in \( x = -1 \) there is a local maximum point.
We will use examples – an example accompanied by Figure 3.4
- Original function: ‘f(x)=x^2'
- First derivative: ‘f'(x)=2x'
- Second derivative: ‘f”(x)=2'
This means that when ‘f”(x)=2', the slope of the original function increases by 2 units per step.
- where x=0, the slope is 0.
- where x = 1, the slope is 2.
- where x = 2, the slope is 4.
- where x = 3, the slope is 6.
Even in the domain where the function decreases (negative gradient), the slopes increase by 2 steps per step. For example:
- where x = 3, the slope is -6.
- where x = -2, the slope is -4.
- where x = -1, the slope is -2.
- where x=0, the slope is 0.
Example with Diagram 3.5
- Original function: ‘f(x)=-x^2'
- First derivative: ‘f'(x)=-2x'
- Second derivative: ‘f”(x)=-2'
This means that the slopes of the original function are reduced by 2 per step.
The information obtained from the second derivative – expansion
When the second derivative is positive at any value of x, then the original function is convex at the same value as x. Also, when the second derivative is positive along the length of any segment, then the original function is convex at the same segment.
When the second derivative is negative at any value of x, then the original function is concave at the same value as x. Also, when the second derivative is negative along a segment, then the original function is concave in the same segment.
The Practical Use of the Second Derivative
background
With the help of the second derivative, we can determine, even without the use of diagrams, which types of curves a point with a slope of 0 belongs to. This is very important. If it belongs to a convex curve, the point should be a minimum point. If it belongs to a concave curve, the point should be a maximum point. If it is in the middle of 2 types of curves, it is an inflection point. The point where the slope 0 is called the 0 point.
Examination of Affiliation, and Conclusions
If the second derivative is positive at the 2 points near the 0 point on either side, the 0 point belongs to a convex curve and is therefore a minimum point. If the second derivative is negative at the 2 points near the 0 point, then the 0 point belongs to a concave curve and is therefore a maximum point. If the second derivative at one adjacent point is positive and the other is negative, the 0 point is between 2 types of curves and is therefore a twisting point.
Illustrative Examples
Only by means of examples can the contribution of the second derivative be gradually digested.
Example 1
- The example refers to an original function with the form ‘f(x)=x^2-6x'.
- Let's assume that we don't know its route.
- Using the first derivative, we can verify whether there are points in the path of the function where the slope = 0 and where (in what values of x are they located). The form of the derivative is: ‘f'(x)=2x-6'. A slope of 0 is obtained only when x=3. That is, where 0=2x-6 and the result is 3=x. and the meaning, where x=3 the slope of the function's path is 0.
- Since we are not familiar with the route of the function, we do not know whether it is a maximum point or a minimum point or a turning point.
- To ascertain which of the 3 options is correct, we need to find the second derivative. The second derivative form is ‘f”(x)=2'. From the result, we learn that the second derivative is positive (2) along the entire length of the original function, which means that the gradients in the original function increase by 2 with each step. It is clear from here that even at the 2 points near the zero point, the second derivative is positive and equal to 2.
Suppose the adjacent dots refer to x values of 2.9 and 3.1. In these two values, the second derivative is positive, and therefore the 0 point is a minimal extreme point.
Example 2
Original function: ‘f(x)=4+5x'
First derivative: ‘f'(x)=5'
In this example, there is no point in the function where the slope is 0. Of course, we know that this is a function that represents a straight line with a slope of 5 along its entire length and therefore does not have a point with a slope of 0.
Example 3
- Original function: ‘f(x)=-3x^2-2x'
- First derivative: ‘f'(x)=-6x-2'
- Second derivative: ‘f”(x)=-6'
From the result we learn that the second derivative is negative along the entire length of the original function, which means that the slopes in the original function decrease by 6 with each step. It is clear from here that even at the 2 points near the zero point, the second derivative is negative and hence it is amaximum extreme point.
Useful and interesting examples of a second derivative
There are quite a few useful and interesting examples of a second derivative, especially in fields such as physics, economics, and engineering. Here are some examples that can illustrate the power of the second derivative:
1. Physics – Acceleration
One of the simplest and clearest examples is the use of the second derivative in the field of physics. If we define the position of an object as a function of time, the first derivative of the position is velocity, and the second derivative is acceleration.
2. Economics – convex analysis for profits or costs
In economics, the second derivative can be used to understand the profit or cost behavior of a company. If the profit or cost function is convex downward (the second derivative is negative), it may indicate that there is a maximum point, which is the maximum profit point or minimum cost.
3. Engineering – Building Design
In engineering structures, it is very important to use the second derivative to analyze forces and loads on beams. In these analyses, the concept of the curvature of structures is used, which provides information about points where the beams may bend or break. A second derivative helps to understand the nature of the force exerted on the structure and how it changes, in order to plan the strength of the structure and ensure durability over time.
4. Optics – Focus of Lenses
In optical systems, the use of a second derivative helps determine the focus point of lenses. A second derivative of a function that describes the passage of light through the lens helps to calculate the point of twist at which the light is concentrated. In optical systems such as telescopes and microscopes, understanding focus points is essential for achieving optimal sharpness.
5. Economics – Market Analysis: Rise and Fall in the Rate of Demand and Supply
In economics, if we examine the demand for a particular product over time, the first derivative describes the rate of change in demand, while the second derivative describes the change in the rate itself. If the second derivative is positive, the rate of demand increases; If it is negative, the rate of demand decreases. In this way, it is possible to understand the demand trends, and to predict whether demand will continue to rise or whether the upward trend will begin to weaken.
6. Data and Statistics Analysis – Curvature of the Probability Graph
In data analysis, especially when it comes to probability graphs, the second derivative helps to understand the curvature of the density function of a random variable. This allows us to know in which areas the distribution of data is more concentrated and in which areas it “spreads”, which helps in understanding the characteristics of the data, such as averages and variance.
summary
The second derivative is an important tool that helps us analyze the behavior of the function, identify the types of extremes, check the curvature, and identify the twisting points. Understanding the second derivative deepens our understanding of the nature of the function and helps us analyze it more accurately.
