Introduction to Third Derivative Calculator:
Third derivative calculator is an online tool that helps you to find the third derivative of the given function in a few seconds. Our tool is used to evaluate the rate of change of the second derivative function f``(x).
Our 3rd derivative calculator is a beneficial tool for students, teachers, professionals, or researchers for making notes, assignments, or projects without doing manual calculations just click on it and get a solution of any third derivative problem.
What is the Third Derivative?
The third derivative of a function is the rate of change of the rate of change of original function which means derivation of the second derivative f``(x). In mathematical notation, f(x) is a function, and the third derivative, f′′′(x) or dx3/d3f, represents the rate of change of the second derivative, f′′(x).
It is used in various fields like in physics the third derivative is known as "jerk" or "jolt" in kinematics, and in engineering, it is used in the analysis of mechanical systems like vibrations and forces. In mathematics, it is used in Taylor series expansions and the behavior of functions.
How to Find the Third Derivative?
To find the third derivative of a function, the third derivative calculator follows some important rules of differentiation. Let us see an example of a third derivative where all the calculation steps are mentioned for a better understanding of the third derivative concept in steps.
Example:
$$ f(x) \;=\; x^4 + 3x^3 + 2x^2 + x + 5 $$
Solution:
Step 1:
The triple derivative calculator first of all identifies the function so for calculations identify the given function f(x) and see which differential rule is applied in the original function that gives you an accurate solution.
$ f(x) \;=\; x^4 + 3x^3 + 2x^2 + x + 5 $$
Step 2:
After identifying the type of function f(x) we use the power rule formula for derivation.
$$ \frac{d}{dx}x^n \;=\; nx^{x-1} $$
Step 3:
To find the third derivative you need to compute the first derivative of the given function f(x).
$$ f’(x) \;=\; \frac{d}{dx}(x^4) + \frac{d}{dx}(3x^3) + \frac{d}{dx}(2x^2) + \frac{d}{dx}(x) + \frac{d}{dx}(5) $$
$$ f’(x) \;=\; 4x^3 + 9x^2 + 4x + 1 $$
Step 4: Again differentiate the function f`(x) with respect to x.
$$ f’’(x) \;=\; \frac{d}{dx}(4x^3) + \frac{d}{dx}(9x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(1) $$
$$ f’’(x) \;=\; 12x^2 + 18x + 4 $$
Step 5: For the Third Derivative of function f(x), you need to find the derivative of f``(x) again with respect to x.
$$ f’’’(x) \;=\; \frac{d}{dx}(12x^2) + \frac{d}{dx}(18x) + \frac{d}{dx}(4) $$
$$ f’’’(x) \;=\; 24x + 18 $$
Therefore the solution of the given example of third derivative is,
$$ f’’’(x) \;=\; 24x + 18 $$
How to Use Third Derivative Calculator?
The triple derivative calculator has a simple layout that helps you to solve the given third derivative question immediately.
You just need to put the input value in this 3rd derivative calculator. Follow some important steps to get a solution without any inconvenience. These steps are:
- Enter the differential function that you want to evaluate the third derivative problem in the input fields.
- Choose the variable of differentiation from the given list.
- Check your given input function to get the correct solution of the third derivation question.
- Click on the Calculate button to get the result of the given third derivation function problems.
- If you want to understand the calculation process of our third order derivative calculator then use the load example and get its solution for more clarity.
- Click the “Recalculate” button for the calculation of more examples of third derivate functions with the solution.
Outcome from 3rd Derivative Calculator:
Third Derivative Calculator provides you with a solution as per your input problem when you click on the calculate button. It may contain as:
Click on the result button so you get the solution of your third derivative function question.
- Steps Box:
When you click on the steps option, you get the result of third differential questions in a step-by-step process
Why to Choose Triple Derivative Calculator:
The third order derivative calculator has many useful features that you get when you use it to solve third derivative problems to get its solution. Our tool only takes the differential function and it provides you a solution without getting any external effort. These features are
- It is a reliable tool because it always provides you with precise solutions to given third-derivative problems.
- It is a speed tool that provides solutions to the given third derivative problems in a few seconds.
- It is a learning tool that gives you with conceptual clarity of third derivative function very easily through online platforms by sitting at home.
- 3rd derivative calculator is a handy tool that can solve various types of third derivation problems quickly and easily.
- It is a free tool that allows you to use it for the calculation of differentiation of functions three times without spending.
- Third derivative calculator is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of third derivation problems without any trouble.
What is the third derivative of sin 2x
To find the third derivative of the function sin(2x), you need to apply the rules of differentiation. Let's find its solution in detail. The given function is,
$$ f(x) \;=\; sin (2x) $$
Use chain rule for differentiation, then
$$ \frac{d}{dx} g(h(x)) \;=\; g’(h(x)) . h’(x) $$
Differentiate the given function f(x) with respect to x.
$$ f’(x) \;=\; \frac{d}{dx} sin(2x) \;=\; cos(2x) \;=\; cos(2x) . 2 \;=\; 2 cos(2x) $$
Again differentiate the f`(x) with respect to x,
$$ f’’(x) \;=\; \frac{d}{dx} [2 cos(2x)] \;=\; 2 . \frac{d}{dx} cos(2x) \;=\; 2 . (-sin(2x)) . \frac{d}{dx} (2x) \;=\; 2 . (-sin(2x)) . 2 $$
For the solution of the third derivation, again take the derivative of f``(x) second order derivative function,
$$ f’’’(x) \;=\; \frac{d}{dx} -4 sin(2x) = -4 . \frac{d}{dx} sin(2x) = -4 . cos(2x) . \frac{d}{dx} (2x) \;=\; -4 . cos(2x) . 2 $$
Hence the solution of the given third derivative function is,
$$ f’’’(x) \;=\; -8 cos(2x) $$
What if first second and third derivatives were 0
If the first, second, and third derivatives of a function f(x) are all zero, it expresses a specific situation about the nature of the function f(x). Let's analyze this in steps.
- First Derivative f′(x) = 0:
The function f(x) is a constant function because the slope of the function is zero everywhere. So, f(x) does not change as x changes.
- Second Derivative f′′(x) = 0:
Since f′(x) = 0f'(x) = 0, f′′(x) = 0 says that f′(x) is not only zero at specific points but is zero everywhere so that f(x) is a constant function.
- Third Derivative f′′′(x) = 0
The third derivative being zero is consistent as the first and second derivatives is zero everywhere, further, it confirms that f(x) is a constant function.
Conclusion
If f′(x) = 0, f′′(x) = 0, and f′′′(x) = 0 for all x, then the function f(x) must be a constant function. In other words, f(x) = C where C is a constant.
What is the purpose of a third derivative
The third derivative of a function has important purposes in both theoretical and applied mathematics. It provides insights into the behavior and properties of the function. Here are some of the key purposes and applications of the third derivative:
- Rate of Change of Curvature:
The third derivative f′′′(x) measures the rate at which this curvature is changing. to understand the nature of points of inflection and the dynamic behavior of curves.
In physics, the third derivative is used to study the motion of the position function with respect to time and is known as "jerk" or "jolt". It represents the rate of change of acceleration.
The third derivative plays a role in the Taylor series expansion of a function.
- Study of Oscillatory Behavior:
In the analysis of oscillatory systems, such as electrical circuits or mechanical vibrations, the third derivative can provide insights into the behavior of the system.
In signal processing, higher-order derivatives, including the third derivative, are used in the design of filters and in the analysis of signals.
- Modeling and Analysis of Financial Data:
In finance, the third derivative can be used in the modeling and analysis of financial data, such as in the study of the sensitivity of option prices.
What is the third derivative of 1/x
To find the third derivative of the function f(x)=1/x, differentiate it three times. For differentiation use the power rule with respect to x,
$$ \frac{1}{x} \;=\; x^{-1} $$
$$ f(x) \;=\; x^{-1} $$
$$ f’(x) \;=\; \frac{d}{dx}(x^{-1}) = -x^{-2} $$
$$ f’(x) \;=\; - \frac{1}{x^2} $$
Again differentiate f`(x) with respect to x,
$$ f’(x) \;=\; -x^2 $$
$$ f’’(x) \;=\; \frac{d}{dx} (-x^2) \;=\; -(-2)x^{-3} $$
$$ f’’(x) \;=\; 2x^{-3} $$
$$ f’’(x) \;=\; \frac{2}{x^3} $$
To get the solution of the third derivative solution, differentiate the second derivative function f``(x) with respect to x.
$$ f’’(x) \;=\; 2x^{-3} $$
$$ f’’’(x) \;=\; \frac{d}{dx}(2x^{-3}) \;=\; 2 . (-3)x^{-4} $$
$$ f’’’(x) \;=\; - 6x^{-4} $$
$$ f’’’(x) \;=\; - \frac{6}{x^4} $$
Therefore the solution of the given third derivative function 1/x is,
$$ f’’’(x) \;=\; - \frac{6}{x^4} $$
What is the third derivative of tan inverse x
To find the third derivative of the function f(x)=tan^−1(x), differentiate it by using the appropriate function three times.
Differentiate f(x) with respect to x.
The given function tan^-1x has the direct formula of differentiation,
$$ f’(x) \;=\; \frac{d}{dx} tan^{-1} (x) \;=\; \frac{1}{1+x^2} $$
Again differentiate f`(x) with respect to x,
$$ f’(x) \;=\; \frac{1}{1+x^2} $$
Use the chain rule or power rule for differentiation,
$$ f’(x) \;=\; (1 + x^2)^{-1} $$
$$ f’’(x) \;=\; \frac{d}{dx} (1 + x^2)^{-1} \;=\; -1 . (1 + x^2)^{-2} . \frac{d}{dx} (1 + x^2) $$
$$ f’’(x) \;=\; -(1 + x^2)^{-2} . 2x $$
$$ f’’(x) \;=\; - \frac{2x}{(1+x^2)^2} $$
Again differentiate f``(x) with respect to x to get the solution of the third derivative function,
$$ f’’(x) \;=\; -2x(1 + x^2)^{-2} $$
Use the product rule or chain rule for derivation,
$$ f’’’(x) \;=\; -2\left[(1 + x^2)^{-2} + x . \frac{d}{dx](1 + x^2)^{-2} \right] $$
$$ f’’’(x) \;=\; -2 \left[(1 + x^2)^{-2} + x . (-4x(1 + x^2)^{-3}) \right] $$
$$ f’’’(x) \;=\; -2[(1 + x^2)^{-2} - 4x^2(1 + x^2)^{-3}] $$
Factor out (1 + x2)-3:
$$ f’’’(x) \;=\; -2 [(1 + x^2)^{-2} (1 + x^2) . (1 + x^2)^{-1} - 4x^2(1 + x^2)^{-3}] $$
$$ f’’’(x) \;=\; -2 [(1 + x^2 - 4x^2)(1 + x^2)^{-3}] $$
$$ f’’’(x) \;=\; -2[(1 - 3x^2)(1 + x^2)^{-3}] $$
$$ f’’’(x) \;=\; -2 \frac{1 - 3x^2}{(1 + x^2)^3} $$
Therefore the solution of the given third derivative function tan^-1x is,
$$ f’’’(x) \;=\; -2 \frac{1 - 3x^2}{(1 + x^2)^3} $$
