# Third Derivative Calculator

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## Third Derivative Calculator

Find the third derivative of any function with this calculator. Also, learn the process of computing the derivatives by reading the solved steps.

## How to use this tool?

The third derivative calculator is pretty straightforward. It requires three steps.

1. Enter the function.
2. Choose the variable.
3. Click Calculate

It automatically calculates the first and second derivatives for the inputted function.

## What is the third derivative?

Before defining the third derivative, it is important to know its background context.

At its core, a derivative represents the rate at which a function's value changes as its input changes. The first derivative, f′(x), measures the slope of the tangent line to the function at any point, providing insight into the function's rate of change.

The second derivative, f′′(x), is crucial for understanding a function's curvature. It tells us how the slope of a function changes, indicating whether a function is concave up or down at a given point.

The third derivative, f′′′(x), is the rate of change of the rate of change of the rate of change of a function. In simpler terms, it measures how the curvature of a function changes.

## The formula for the third derivative:

The formula for the third derivative of a function, in its most general form, is derived by applying the process of differentiation three times to the function.

In mathematical terms, if f(x) is a function, the third derivative f′′′(x) is calculated as:

$f'''(x) = \frac{d}{dx}\left(\frac{d^2f}{dx^2}\right)$

This formula represents the continuous application of the differentiation process, each step building on the previous one. The third derivative can be more complex to compute, especially for functions that are not straightforward.

However, it follows the fundamental principles of differentiation, applying the same rules and methods used for finding the first derivative and second derivatives.

## How to calculate the third derivative?

A walk-through of the computation process of the third derivative can help you understand how to do it manually better.

Example:

Let's take a moderately complex function and find its third derivative. We'll use the function $$f(x) = x^3e^x$$ and apply the product rule for differentiation.

Solution:

First Derivative:

To find the first derivative f′(x), apply the product rule, which states: $(uv)’ = u’v + uv’$

where, \begin{align*}
u &= x^3 & \text{and} \quad v &= e^x
\end{align*}Put the values of above then we obtain
\begin{align*}f'(x) &= (x^3)' e^x + x^3 (e^x)' \\
&= 3x^2 e^x + x^3 e^x \\
&= e^x (3x^2 + x^3)
\end{align*}

Second Derivative:

Next, differentiate f′(x) to get f′′(x).

$f''(x) = \frac{d}{dx}(e^x (3x^2 + x^3))$

Use product rule again:

\begin{align*}
\frac{d}{dx}(e^x (3x^2 + x^3)) &= e^x \frac{d}{dx}(3x^2 + x^3) + (3x^2 + x^3) \frac{d}{dx}(e^x) \\
&= e^x (6x + 3x^2) + (3x^2 + x^3) e^x \\
&= e^x (6x + 6x^2 + x^3)
\end{align*}

Third derivative:

Finally, differentiate f′′(x) to find f′′′(x).

\begin{align*}
[f'''(x) &= \frac{d}{dx}(e^x (6x + 6x^2 + x^3))]
\end{align*}

Again, use the product rule:

\begin{align*}
&= e^x \frac{d}{dx}(6x + 6x^2 + x^3) + (6x + 6x^2 + x^3) \frac{d}{dx}(e^x) \\
&= e^x (6 + 12x + 3x^2) + (6x + 6x^2 + x^3) e^x \\
&= e^x (6 + 12x + 3x^2 + 6x + 6x^2 + x^3) \\
&= e^x (6 + 18x + 9x^2 + x^3)
\end{align*}

Conclusion:

The third derivative of $$f(x) = x^3e^x$$ is:

$f'''(x) = e^x (6 + 18x + 9x^2 + x^3)$

## Applications:

Physics - 'Jerk' and Motion: In physics, the third derivative is known as 'jerk', the rate of change of acceleration. It's important to understand how an object's motion changes in dynamics.

Economics: Economists use the third derivative to analyze changes in rates such as inflation or growth, providing deeper insights into economic trends.

Engineering: In material science, the third derivative helps in understanding stress-strain relationships in materials under various forces.

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