A derivative is a well-known branch of calculus that is widely used to evaluate the slope of the tangent line and the rate of change in the functions. The first, second, and third derivatives are the advanced form of differentiation.
The first, second, and third derivatives are the fundamental concept of calculus that deals with advanced calculus functions and calculations. In this article, we’ll learn the first, second, and third derivatives with examples.
Table of Contents
What is the first derivative?
In calculus, the instantaneous rate of change of a function with respect to an independent variable to evaluate the slope of curves is said to be the first derivative. The derivative of a function is widely used to give the slope of the tangent line to the graph of the original function at any point.
The process of evaluating the differential of polynomial, exponential, logarithmic, and trigonometric functions is known as differentiation. The first derivative is denoted by f’(t) or d/dt. The formula for the first derivative with the help of limit is:
d/dt f(t) = limh→0 [f(t + h) – f(t)] / h
What is the second derivative?
In calculus, the derivative of a derivative is said to be the second derivative. Such as the derivative of the 1st derivative of a function is known as the 2nd derivative. The curvature of a graph can be represented with the help of the second derivative of the function.
It is denoted by f’’(t), d/dt (d/dt), or d2/dt2. There are three possibilities for the 2nd differential function:
Concave up
If the 2nd derivative is positive at a given point, then the function is said to be concave up at that point, meaning that the graph is shaped like a cup, with the tangent line sloping upwards.
Concave down
If the 2nd derivative is negative, then the function is concave down, meaning that the graph is shaped like a bowl, with the tangent line sloping downwards.
Inflection point
If the second 2nd is zero, then the function is said to have an inflection point, where the curvature of the graph changes from concave up to concave down or concave down to concave up.
What is the third derivative?
In calculus, the derivative of a derivative of a derivative is referred to as the 3rd derivative of a function. in simple words, the derivative of the second derivative is known as the third derivative. The working of the 3rd derivative is similar to the 1st and 2nd derivatives but it is less commonly used than the other two.
The third derivative is represented by f’’’(t), d/dt [d/dt (d/dt)], or d3/dt3. The concept of the third derivative can be extended to higher-order derivatives, such as the fourth derivative, the fifth derivative, and so on.
Higher-order derivatives can be used to analyze the behavior of functions in even more detail, but their practical applications are relatively rare compared to the first and second derivatives.
Rules of differentiation
Here are some rules of differentiation.
| Rules name | First Derivative | 2nd Derivative | 3rd Derivative |
| Trigonometry Rule | d/dv [sin(v)] = cos(v) d/dv [cos(v)] = -sin(v) d/dv [tan(v)] = sec2(v) | d2/dv2 [sin(v)] = -sin(v) d2/dv2 [cos(v)] = -cos(v) d2/dv2 [tan(v)] = 2sec2(v) tan(v) | d3/dv3 [sin(v)] = -cos(v) d3/dv3 [cos(v)] = sin(v) d2/dv2 [tan(v)] = -4sec2(v) + 6sec4(v) |
| Power Rule | d/dv [f(v)]n = n [f(v)]n-1 * [f’(v)] | d2/dv2 [f(v)]n = n (n – 1) [f(v)]n-2 * [f’’(v)] | d3/dv3 [f(v)]n = n (n – 1) (n – 2) [f(v)]n-3 |
| Quotient Rule | d/dv [f(v) / g(v)] = 1/[g(v)]2 [g(v) * [f’(v)] – f(v) * [g’(v)]] | d2/dv2 [f(v) / g(v)] | d3/dv3 [f(v) / g(v)] |
| Product Rule | d/dv [f(v) * g(v)] = g(v) * [f’(v)] + f(v) * [g’(v)] | d2/dv2 [f(v) * g(v)] | d3/dv3 [f(v) * g(v)] |
| Constant Rule | d/dv [L] = 0, where L is any constant | d2/dv2 [L] = 0, where L is any constant | d3/dv3 [L] = 0, where L is any constant |
| Difference Rule | d/dv [f(v) – g(v)] = [f’(v)] – [g’(v)] | d2/dv2 [f(v) – g(v)] = [f’’(v)] – [g’’(v)] | d3/dv3 [f(v) – g(v)] = [f’’’(v)] – [g’’’(v)] |
| Sum Rule | d/dv [f(v) + g(v)] = [f’(v)] + [g’(v)] | d2/dv2 [f(v) + g(v)] = [f’’(v)] + [g’’(v)] | d3/dv3 [f(v) + g(v)] = [f’’’(v)] + [g’’’(v)] |
| Exponential Rule | d/dv [ev] = ev d/dv [v] | d2/dv2 [ev] = ev d2/dv2 [v] | d3/dv3 [ev] = ev d3/dv3 [v] |
| Constant function Rule | d/dv [L * f(v)] = L [f’(v)] | d2/dv2 [L * f(v)] = L [f’’(v)] | d3/dv3 [L * f(v)] = L [f’’’(v)] |
How to calculate 1st, 2nd, & 3rd derivatives?
The 1st, 2nd, & 3rd derivatives can be evaluated with the help of rules of differentiation. Alternatively, a differential calculator can be used to evaluate derivative problems with steps to get the result in few seconds.
Here is an example for 1st, 2nd, & 3rd derivatives to understand how to evaluate them manually.
Example
Evaluate 1st, 2nd, & 3rd derivatives of the given algebraic expression with respect to “v”.
f(v) = 2v3 + 15v2 – 4v5 + 12cos(v) + 6v6
Solution
For First Derivative
Step 1: First of all, apply the notation of the derivative to the given function.
d/dv f(v) = d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]
Step 2: Now apply the sum and difference rules of differentiation to the above expression and take out constant coefficients.
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = d/dv [2v3] + d/dv [15v2] – d/dv [4v5] + d/dv [12cos(v)] + d/dv [6v6]
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 2d/dv [v3] + 15d/dv [v2] – 4d/dv [v5] + 12d/dv [cos(v)] + 6d/dv [v6]
Step 3: Now use the power rule of differentiation in the above expression.
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 2 [3 v3-1] + 15 [2 v2-1] – 4 [5 v5-1] + 12 [-sin(v)] + 6 [6 v6-1]
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 2 [3 v2] + 15 [2 v1] – 4 [5 v4] + 12 [-sin(v)] + 6 [6 v5]
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 2 [3 v2] + 15 [2 v] – 4 [5 v4] – 12 [sin(v)] + 6 [6 v5]
d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 6v2 + 30v – 20v4 – 12sin(v) + 36v5
For Second Derivative
Step 1: First of all, apply the notation of the derivative to the first derivative of the function.
d/dv [d/dv [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = d/dv [6v2 + 30v – 20v4 – 12sin(v) + 36v5]
Step 2: Now apply the sum and difference rules of differentiation to the above expression and take out constant coefficients.
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = d/dv [6v2] + d/dv [30v] – d/dv [20v4] – d/dv [12sin(v)] + d/dv [36v5]
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 6d/dv [v2] + 30d/dv [v] – 20d/dv [v4] – 12d/dv [sin(v)] + 36d/dv [v5]
Step 3: Now use the power rule of differentiation in the above expression.
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 6 [2 v2-1] + 30 [v1-1] – 20 [4 v4-1] – 12 [cos(v)] + 36 [5 v5-1]
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 6 [2 v1] + 30 [v0] – 20 [4 v3] – 12 [cos(v)] + 36 [5 v4]
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 6 [2 v] + 30 [1] – 20 [4 v3] – 12 [cos(v)] + 36 [5 v4]
d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6] = 12v + 30 – 80v3 – 12cos(v) + 180 v4
For Third Derivative
Step 1: First of all, apply the notation of the derivative to the second derivative of the function.
d/dv [d2/dv2 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = d/dv [12v + 30 – 80v3 – 12cos(v) + 180v4]
Step 2: Now apply the sum and difference rules of differentiation to the above expression and take out constant coefficients.
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = d/dv [12v] + d/dv [30] – d/dv [80v3] – d/dv [12cos(v)] + d/dv [180v4]
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = 12d/dv [v] + d/dv [30] – 80d/dv [v3] – 12d/dv [cos(v)] + 180d/dv [v4]
Step 3: Now use power and constant rule of differentiation in the above expression.
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = 12 [v1-1] + [0] – 80 [3 v3-1] – 12 [-sin(v)] + 180 [4 v4-1]
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = 12 [v0] + [0] – 80 [3 v2] – 12 [-sin(v)] + 180 [4 v3]
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = 12 [1] + [0] – 80 [3 v2] + 12 [sin(v)] + 180 [4 v3]
d3/dv3 [2v3 + 15v2 – 4v5 + 12cos(v) + 6v6]] = 12 – 240v2 + 12sin(v) + 720v3
Hence, the first, second, and third derivatives of the given function are:
| 1st derivative | 2nd derivative | 3rd derivative |
| 6v2 + 30v – 20v4 – 12sin(v) + 36v5 | 12v + 30 – 80v3 – 12cos(v) + 180 v4 | 12 – 240v2 + 12sin(v) + 720v3 |
Wrap Up
The first, second, and third derivatives are the advanced form of differentiation to evaluate the rate of change. Now you can grab the basics of these advanced differentiation techniques from this post.