Calculated Result
Derivative Entered
Antiderivative
Rule Used
Formula Used
Steps
Verification Table
| Sample x | Original Derivative | Estimated d/dx of Result | Difference |
|---|
Calculator
Formula Used
If the derivative is a·xn, the original function is a/(n+1) · xn+1 + C, where n ≠ -1.
If the derivative is a/(kx+b), the original function is (a/k) ln|kx+b| + C.
If the derivative is a·e^(kx+b), the original function is (a/k)e^(kx+b) + C, when k ≠ 0.
If the derivative is a·m^(kx+b), the original function is a / (k ln m) · m^(kx+b) + C.
Selected trigonometric rules are also included. For example, a·cos(kx+b) becomes (a/k)sin(kx+b) + C.
How to Use This Calculator
- Choose a single pattern or polynomial series mode.
- Select the derivative family when single mode is active.
- Enter the needed coefficients, exponents, or inner values.
- Set the variable symbol and your preferred decimal places.
- Click the calculate button to show the result above the form.
- Review the rule, formula, steps, and verification table.
- Download the current result as CSV or PDF if needed.
Example Data Table
| Derivative | Rule Family | Original Function | Note |
|---|---|---|---|
| 6x² | Power rule | 2x³ + C | Divide by the new exponent. |
| 3 / (2x + 1) | Reciprocal rule | 1.5 ln|2x + 1| + C | Divide by the inner coefficient. |
| 4e^(3x) | Natural exponential | (4/3)e^(3x) + C | Reverse the chain rule. |
| 5cos(2x) | Trigonometric | 2.5sin(2x) + C | Cos integrates to sine. |
| 7 | Constant rule | 7x + C | A constant becomes linear. |
Inverse Differentiation Guide
Understanding Inverse Differentiation
Inverse differentiation means finding an original function from its derivative. Many students call this reverse differentiation. In calculus, it is integration. This calculator focuses on common derivative patterns. It gives a fast original function. It also keeps the constant of integration visible. That part matters in every antiderivative problem.
What This Calculator Does
The tool handles several useful cases. You can work with power terms, constants, reciprocal forms, exponential expressions, and selected trigonometric derivatives. It also supports a polynomial series mode. That helps when a derivative has many terms. Instead of solving each term by hand, you can reverse the rule step by step. The result section appears above the form. It shows the derivative, the antiderivative, the rule used, and a small verification table.
Why the Rules Work
If a derivative looks like axn, the original function is a divided by n plus one, then x raised to n plus one. The only exception is n equals negative one. That form becomes a logarithm. Similar logic applies to exponential and trigonometric cases. A matching inner coefficient changes the final answer. You must divide by that inner rate. This is the reverse of the chain rule. Many errors happen when students forget that adjustment.
How the Verification Helps
The calculator also compares values numerically. It estimates the derivative of the produced antiderivative. Then it compares that estimate with the derivative you entered. Small differences can appear from rounding. That is normal. The check still helps you trust the answer. It is also useful for practice, homework review, and quick classroom examples.
Where This Tool Is Useful
Use it when you want cleaner calculus practice. Use it when you need a reliable check before writing a final solution. It is helpful for revision sheets, lesson planning, and tutoring notes. The export buttons also help. You can save the current result as CSV or PDF. That makes record keeping easier. A simple example table is included below. It gives a quick view of common reverse differentiation outputs.
Each family has limits. The calculator explains those limits clearly. When a power exponent is negative one, switch to the reciprocal rule. That guidance reduces confusion and prevents incorrect algebra during practice.
FAQs
What does inverse differentiation mean?
It means finding an original function from a derivative. In calculus, that process is integration or antidifferentiation.
Does this tool solve every antiderivative?
No. It focuses on common patterns such as power, reciprocal, exponential, constant, and selected trigonometric derivatives.
Why is the constant of integration included?
Different functions can share the same derivative. The added constant shows the full family of original functions.
When should I use the reciprocal option?
Use it when the derivative looks like a number divided by a linear expression, such as 3 divided by 2x plus 1.
What happens if the power exponent is negative one?
The standard power rule does not apply. That case becomes a logarithmic antiderivative, so use the reciprocal family instead.
Can I enter several polynomial terms together?
Yes. Use polynomial series mode and supply matching coefficient and exponent lists in the same order.
How does the verification table help?
It compares your derivative with a numerical derivative of the produced antiderivative at sample x values.
What do the CSV and PDF buttons export?
They export the current result summary and the verification rows shown after calculation.