Finding Derivative, Second Derivative, and Curvature
By Kathleen Knowles, 03 Jul 2020
The mathematical study of calculus requires a deep understanding of a fundamental concept: derivatives. For some, the idea of derivatives in calculus comes naturally; it becomes an intriguing idea with countless applications to understanding the real world.
For others, it’s merely a source of confusion or unnecessary stress. Due to their fundamental application to calculus, a misunderstanding of derivatives can also lead to unnecessarily lower grades and stressed students.
When read properly, this article can alleviate some of your concerns with a proper explanation of derivatives and their applications. It is important to note that these are general overviews, and watching video examples on specific rules or methods can allow you to apply what you’ve learned more efficiently.
Finding the Derivative
Before finding the derivative, it will be helpful to define and thoroughly understand what a derivative is. Simply put, the derivative is the slope. More specifically, it is the slope of the tangent line at a given point in a function. To make this more understandable, let’s look at the function f(x) = x^2 at the point (1, 1) on a graphing calculator.
The function is graphed as a U-shaped parabola, and at the point where x=1, we can draw a tangent line. This means that at (1, 1), we can draw a line that touches only this point and is below the curve on either side of this same point. The slope of this line (which is 2) is actually the derivative at that given point.
To actually find this value of the derivative, there are two main methods.
1. The Definition of the Derivative
This method relates to a conceptual understanding of the derivative. If you let the x-axis difference between two points on a curve equal h, this definition of the derivative can be derived and explained in further detail. For the purposes of this explanation, things are simplified with a statement of the formula.
f’(x) = dy/dx = lim as h→0 of [f(x+h) - f(x)] / h
It is really a representation of 'rise over run' or the slope between two points, where the x-axis value between the two points is a, and the distance between the two points is approaching 0.
Example: Find the derivative of f(x) = x^2 at (3, 9).
f’(3) = dy/dx= lim as h→0 of [f(3+h) - f(3)] / h = lim as h→0 of [(3+h)^2 - 9] / h
f’(3) = (9 + 6h + h^2 - 9) / h = 6
2. General Rules
This method is a lot more methodical, and can be used more generally to find the slope at any given point. It does, however, require understanding of several different rules which are listed below.
The Power Rule
dy/dx = nx^(n-1)
This rule finds the derivative of an exponential function.
Example: dy/dx = 4x^3
The Product Rule
dy/dx[f(x) x g(x)] = f(x)g’(x) + f’(x)g(x)
This rule finds the derivative of two multiplied functions.
Example: dy/dx[(3x^2)(x^4)] = (3x^2)(4x^3) + (x^4)(6x) = 12x^5 + 6x^5
The Quotient Rule
dy/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2
This rule finds the derivative of divided functions.
Example: dy/dx = [(3x^2)(4x^3)-(x^4)(6x)]/(3x2)^2 = (2x^5)/(3x^4)
The Chain Rule
dy/dx[(f(g(x))] = f’(g(x))g’(x)
This rule finds the derivative of two functions where one is within the other. It is frequently forgotten and takes practice and consciousness to remember to add it on.
Example: f(x) = x^4 g(x) = 3x^2
dy/dx[(f(g(x))] = f’(3x^2) x 6x
The Constant Multiple Rule
dy/dx[cx^n] = cnx^(n-1)
The derivative of the constant multiple is always just the constant multiple.
Example: dy/dx[6x^3] = 6 x 3x^2 = 18x^2
The Second Derivative
After establishing how to find the first derivative, the second derivative comes fairly easily. The second derivative is simply the derivative of that initial derivative. For example, if the function was f(x), and you took the derivative, the first derivative would be f’(x). The second derivative would be the derivative of f’(x), and it would be written as f’’(x).
Curvature can actually be determined through the use of the second derivative.
When the second derivative is a positive number, the curvature of the graph is concave up, or in a u-shape. When the second derivative is a negative number, the curvature of the graph is concave down or in an n-shape.
It’s easier to understand this through an example. The applications of derivatives are often seen through physics, and as such, considering a function as a model of distance or displacement can be extremely helpful.
If there is a function graphing the distance of a car in meters over time in seconds, the speed of the car is going to be distance over time or the slope of that function at any given point. You will come to realize that the speed of this car is essentially the first derivative.
Now let’s consider the second derivative. Since the first derivative models how fast the function is changing, the second derivative models how fast the first derivative is changing.
An easier way to look at this is with the acceleration of the car. The acceleration of the car shows how fast the speed or the first derivative of the car is changing. Acceleration is, therefore, a good example of the second derivative.
When acceleration is positive, this means that the speed at which the car is increasing speed is increasing. On a graph of the distance, this appears in the u-shape, which we can describe as the concave up curvature.
When acceleration is negative, this means that the speed at which the car is increasing speed is decreasing. On a graph representing the distance traveled, this would instead appear as an n-shape, which represents the concave down curvature.
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