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## Third derivative calculator

This third derivative calculator is designed to calculate the 3rd-order differentiation of any function efficiently. By offering step-by-step guidance and precise results, the calculator makes it easy for users to gain insights into the change in curvature of functions.

At the bottom of the tool, users can find three columns of examples that range from easy to medium to hard which help to understand how the third derivative reveals changes in a function's curvature, making the calculus studies even more engaging and enjoyable.

## How to use this calculator?

The instructions to use the 3rd derivative calculator are as follows.

1. Enter the function in the designated box. Take help from the math keys
2. Choose the w.r.t variable.
3. Confirm the function and click calculate.

## What is the third derivative?

The third derivative provides insights into the behavior of functions beyond the first and second derivatives. As an extension of the concepts of rate of change, the third derivative allows us to analyze a function's "change in curvature" or "jerk".

It is the result of taking the derivative of a function three times. In other words, it represents the rate of change of the second derivative, which in turn represents the rate of change of the first derivative, the original rate of change of the function. The third derivative is often denoted as f'''(x) or d³f(x)/dx³.

## How to find the third derivative?

The third derivative formula is derived by taking the derivative of a function three times. No specific formula for the third derivative applies to all functions, as the third derivative will depend on the original function itself.

However, the process of finding the third derivative involves applying basic differentiation rules repeatedly. Let's go through this process step-by-step using general rules of differentiation:

2. Apply the appropriate differentiation rule(s) to obtain the first derivative, f'(x).
3. Apply the differentiation rule(s) again to the first derivative to obtain the second derivative, f''(x).
4. Finally, apply the differentiation rule(s) to the second derivative to obtain the third derivative, f'''(x) or d³f(x)/dx³.

### Example 1 (Easy):

Find the third derivative of f(x) = x^3.

Solution

Step 1: Calculate the first derivative.

f'(x) = 3x^2 (using the power rule)

Step 2: Calculate the second derivative.

f''(x) = 6x (using the power rule again)

Step 3: Calculate the third derivative.

f'''(x) = 6 (using the power rule one more time)

### Example 2 (Medium):

Find the third derivative of f(x) = (x^2 + 1)e^x.

Solution

Step 1: Calculate the first derivative (using the product rule).

f'(x) = (2x)e^x + (x^2 + 1)e^x

Step 2: Calculate the second derivative (using the product rule again).

f''(x) = (2)e^x + (2x)e^x + (2x)e^x + (x^2 + 1)e^x

f''(x) = (2)e^x + (4x)e^x + (x^2 + 1)e^x

Step 3: Calculate the third derivative (using the product rule again).

f'''(x) = (4)e^x + (4)e^x + (4x)e^x + (2x)e^x + (x^2 + 1)e^x

f'''(x) = (6)e^x + (6x)e^x + (x^2 + 1)e^x

### Example 3 (Hard):

Find the third derivative of f(x) = (x^2 + 1)sin(x).

Solution

Step 1: Calculate the first derivative (using the product rule).

f'(x) = 2x*sin(x) + (x^2 + 1)cos(x)

Step 2: Calculate the second derivative (using the product rule for both terms).

f''(x) = (2sin(x) + 2xcos(x)) + (-x^2cos(x) - 2xsin(x) + cos(x))

f''(x) = 2sin(x) + 2xcos(x) - x^2cos(x) - 2xsin(x) + cos(x)

Step 3: Calculate the third derivative (using the product rule for each term).

f'''(x) = (2cos(x) - 2cos(x) - 2xsin(x)) + (-2sin(x) - x^2sin(x) - 2sin(x) - cos(x))

f'''(x) = - 2xsin(x) - x^2sin(x) - 3*sin(x) - cos(x)

## Applications of the third derivative

While the third derivative may not be as commonly encountered as the first and second derivatives, it does have practical applications. Some examples include:

Motion Analysis: In physics, the third derivative of position with respect to time is called "jerk." Jerk represents the rate of change of acceleration, which is the second derivative of position. Understanding jerk is important for analyzing the smoothness of motion in mechanical systems, such as vehicles, roller coasters, and elevators. By minimizing jerk, engineers can design systems that provide a smoother and more comfortable experience for passengers.

Material Science: The third derivative can be used to study the mechanical properties of materials, specifically the way they respond to applied forces. By examining the third derivative of a stress-strain curve, researchers can gain insights into the transition points of materials, which can be crucial for designing components that must withstand dynamic loads and stresses in various applications.

Biomechanics: The third derivative is used in biomechanics to study human movement and muscle activity. Analyzing jerk can help researchers understand the efficiency of motion and identify areas where improvements can be made to reduce the risk of injury or enhance performance in sports and rehabilitation.

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