Third Derivative Calculator
Calculate the Third Derivative
Enter the coefficients of your polynomial function (up to x5) and the point 'x' at which you want to evaluate the derivatives.
Results
f(x) at x =
f'(x) at x =
f"(x) at x =
f'(x) =
f"(x) =
f"'(x) =
Derivatives Table and Graph
| Derivative | Expression | Value at x |
|---|---|---|
| f(x) | ||
| f'(x) | ||
| f"(x) | ||
| f"'(x) |
Table showing the function and its first three derivatives as expressions and evaluated at the given x.
Graph of f(x) (blue) and f"'(x) (red) around the point x.
What is the Third Derivative Calculator?
A third derivative calculator is a tool that computes the third derivative of a function, typically a polynomial, with respect to its variable. The third derivative, often denoted as f"'(x) or d³y/dx³, measures the rate of change of the second derivative, which itself measures the rate of change of the first derivative (the slope of the function). In physics, if a function represents position with respect to time, the first derivative is velocity, the second is acceleration, and the third derivative is jerk (or jolt), representing the rate of change of acceleration.
This third derivative calculator allows you to input the coefficients of a polynomial function up to the fifth degree and a specific point 'x'. It then calculates and displays the expressions for the first, second, and third derivatives, as well as their values at the given point 'x'.
Who Should Use It?
Students of calculus, engineers, physicists, and anyone working with rates of change of rates of change can benefit from a third derivative calculator. It's particularly useful for understanding the concept of jerk in motion, or more abstractly, the curvature's rate of change.
Common Misconceptions
A common misconception is that higher-order derivatives beyond the second have no practical meaning. While velocity and acceleration are more intuitive, the third derivative (jerk) is important in fields like engineering (designing smooth rides in elevators or roller coasters) and physics (analyzing non-uniform acceleration).
Third Derivative Formula and Mathematical Explanation
For a polynomial function of the form:
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
The first derivative is found by applying the power rule (d/dx(xk) = kxk-1) to each term:
f'(x) = nanxn-1 + (n-1)an-1xn-2 + … + 2a2x + a1
The second derivative is the derivative of f'(x):
f"(x) = n(n-1)anxn-2 + (n-1)(n-2)an-1xn-3 + … + 2a2
The third derivative is the derivative of f"(x):
f"'(x) = n(n-1)(n-2)anxn-3 + (n-1)(n-2)(n-3)an-1xn-4 + … + 6a3
For our third derivative calculator, which handles up to x5 (f(x) = a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0):
- f'(x) = 5a5*x^4 + 4a4*x^3 + 3a3*x^2 + 2a2*x + a1
- f"(x) = 20a5*x^3 + 12a4*x^2 + 6a3*x + 2a2
- f"'(x) = 60a5*x^2 + 24a4*x + 6a3
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a5, a4, a3, a2, a1, a0 | Coefficients of the polynomial terms | Depends on the context of f(x) | Any real number |
| x | The point at which derivatives are evaluated | Depends on the context of f(x) | Any real number |
| f(x) | Value of the function at x | Depends on the context | Calculated |
| f'(x) | First derivative at x (rate of change) | Units of f(x) / units of x | Calculated |
| f"(x) | Second derivative at x (rate of change of f'(x)) | Units of f(x) / (units of x)² | Calculated |
| f"'(x) | Third derivative at x (rate of change of f"(x)) | Units of f(x) / (units of x)³ | Calculated |
Variables used in the third derivative calculation.
Practical Examples (Real-World Use Cases)
Example 1: Jerk in Vehicle Motion
Suppose the position of a vehicle at time t (in seconds) is given by s(t) = 0.1t3 – 0.5t2 + 2t + 5 meters. We want to find the jerk at t = 3 seconds.
Here, a3=0.1, a2=-0.5, a1=2, a0=5 (and a5=0, a4=0). x is t=3.
- s(t) = 0.1t3 – 0.5t2 + 2t + 5
- s'(t) (velocity) = 0.3t2 – t + 2
- s"(t) (acceleration) = 0.6t – 1
- s"'(t) (jerk) = 0.6
At t=3, the jerk s"'(3) = 0.6 m/s3. This constant jerk means the acceleration is changing at a steady rate. Our third derivative calculator would show this if you input a3=0.1, a2=-0.5, a1=2, a0=5, x=3.
Example 2: Analyzing Function Behavior
Consider the function f(x) = x4 – 6x2 + x. We want to understand the rate of change of concavity at x=1.
a4=1, a3=0, a2=-6, a1=1, a0=0.
- f(x) = x4 – 6x2 + x
- f'(x) = 4x3 – 12x + 1
- f"(x) = 12x2 – 12 (Concavity)
- f"'(x) = 24x (Rate of change of concavity)
At x=1, f"'(1) = 24(1) = 24. This positive value indicates that the concavity (given by f"(x)) is increasing at x=1. Using the third derivative calculator with a4=1, a2=-6, a1=1, x=1 will yield f"'(1)=24.
How to Use This Third Derivative Calculator
- Enter Coefficients: Input the coefficients (a5, a4, a3, a2, a1, a0) for your polynomial function f(x) = a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0. If your polynomial is of a lower degree, enter 0 for the higher-order coefficients. For example, for f(x) = 2x^3 – x + 5, enter a3=2, a1=-1, a0=5, and a5=0, a4=0, a2=0.
- Enter x Value: Input the specific value of 'x' at which you want to evaluate the function and its derivatives.
- View Results: The calculator will automatically update and display the expressions for f'(x), f"(x), f"'(x), and their values at the entered 'x'. The primary result, f"'(x) at x, is highlighted.
- Analyze Table and Graph: The table summarizes the function and its derivatives, while the graph visually represents f(x) and f"'(x) around the point x.
- Reset: Click "Reset" to clear inputs to default values.
- Copy Results: Click "Copy Results" to copy the main results and expressions to your clipboard.
The third derivative calculator provides immediate feedback, helping you understand how changes in coefficients or the 'x' value affect the derivatives.
Key Factors That Affect Third Derivative Results
The value of the third derivative, f"'(x), of a polynomial depends primarily on:
- Coefficients of x3, x4, x5 (a3, a4, a5): The third derivative of terms x2, x, and the constant term is zero. Therefore, only coefficients of x3 and higher powers influence the expression of f"'(x). For f"'(x) = 60a5*x^2 + 24a4*x + 6a3, a3, a4, and a5 directly scale its components.
- The Value of x: If the third derivative f"'(x) is not a constant (i.e., if a4 or a5 are non-zero), its value will change depending on the point 'x' at which it is evaluated.
- The Degree of the Polynomial: If the original polynomial is of degree 2 or less, the third derivative will always be zero. If it's degree 3, the third derivative is a constant. If it's degree 4 or 5, the third derivative is a linear or quadratic function of x, respectively.
- Presence of Higher Order Terms: The more terms with powers of x greater than or equal to 3, the more complex the third derivative expression becomes (up to a quadratic for x^5).
- Signs of Coefficients: The signs of a3, a4, and a5 will determine the sign of the terms in f"'(x) and thus influence whether f"'(x) is positive, negative, or zero at a given x.
- Magnitude of Coefficients: Larger magnitudes of a3, a4, and a5 will lead to larger magnitudes in the value of f"'(x), indicating a more rapid change in f"(x).
Understanding these factors is crucial when using the third derivative calculator for analysis.
Frequently Asked Questions (FAQ)
- Q1: What does the third derivative represent physically?
- A1: If a function represents position over time, the third derivative is "jerk," the rate of change of acceleration. It measures how smoothly or abruptly acceleration changes.
- Q2: Can I use this calculator for non-polynomial functions?
- A2: No, this specific third derivative calculator is designed for polynomial functions up to the 5th degree. For other functions (like trigonometric or exponential), different differentiation rules apply, and you'd need a more general derivative calculator.
- Q3: What if my polynomial is of degree 2 or less?
- A3: The third derivative of a quadratic, linear, or constant function is always zero. The calculator will show f"'(x) = 0.
- Q4: What if the third derivative is zero?
- A4: If f"'(x) = 0 at a point or over an interval, it means the second derivative (e.g., acceleration) is not changing at that point or over that interval.
- Q5: How is the third derivative related to concavity?
- A5: The second derivative f"(x) tells us about the concavity of f(x). The third derivative f"'(x) tells us how the concavity is changing.
- Q6: What does a large third derivative value mean?
- A6: A large f"'(x) (in magnitude) means the second derivative is changing rapidly. In motion, it means acceleration is changing quickly (high jerk).
- Q7: Can the calculator handle negative coefficients or x values?
- A7: Yes, the third derivative calculator accepts negative and decimal values for both the coefficients and the point x.
- Q8: Why does the graph only show a small range around x?
- A8: The graph focuses on the behavior of f(x) and f"'(x) near the specific point x you entered to give a localized view.