Power Rule Derivative Calculator for a·x^n
Differentiate one term a·x^n with the power rule: 5x^4 becomes 20x^3, because the coefficient is multiplied by the exponent and the exponent drops by one. It handles negative, decimal and zero exponents too. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Coefficient
- 5
- Exponent
- 4
- Derivative coefficient
- 20
- Derivative exponent
- 3
The power rule derivative calculator differentiates one term in the form a times x raised to n. Enter the term's numeric coefficient and exponent, and it returns the coefficient and exponent of the derivative. The calculation applies the familiar rule d/dx(a·x^n) = (a·n)·x^(n−1) directly, without symbolic parsing or hidden simplification. It is useful for checking coursework, preparing worked examples, and generating dependable coefficient-exponent pairs in automated mathematics workflows with clear structured output.
Enter the two parts of the power term
Represent the term a·x^n with two numbers: coefficient is a, and exponent is n. For example, 5x^4 has coefficient 5 and exponent 4. The variable itself does not need to be entered because changing its letter does not change the power-rule arithmetic.
Both values must be finite numbers. Integers, decimals, negative values, and zero are accepted, so the calculator can handle examples such as −3x^2, 0.5x^−4, and 7x^0. Keep the input focused on one term: sums such as 3x^2 + 4x are not a single power term and should be differentiated term by term.
Likewise, do not place notation such as “x^4” into either numeric field.
Read the derivative coefficient and exponent
The calculator applies two operations. With coefficient 5 and exponent 4, the returned pair is 20 and 3, representing 20x^3. Negative exponents follow the same rule: 2x^−3 becomes −6x^−4.
Fractional exponents also work numerically; 8x^0.5 produces coefficient 4 and exponent −0.5. A constant term has exponent zero, so multiplication produces a zero derivative coefficient. The returned exponent is still n−1, which is −1 in that mechanical power-rule representation; because the coefficient is zero, the derivative is simply zero.
The output deliberately contains numbers rather than a formatted expression. That structure is easier to copy into another calculation, validate in a test, or serialize without disagreements about notation. It also keeps downstream comparisons precise and independent of display preferences.
Know the scope and numerical limits
This capability is intentionally narrow: it applies the power rule to exactly one numeric monomial and returns the two numeric components of its derivative. It does not combine like terms, parse an equation, choose a variable, evaluate the derivative at a point, or apply product, quotient, chain, logarithmic, or trigonometric rules. Those operations require more information than a coefficient and exponent provide.
Inputs are processed with standard finite JavaScript numbers, so extremely large magnitudes can exceed the representable range. Ordinary classroom, engineering, and spreadsheet-scale values are well within the supported range. The computation has no network calls, random choices, stored state, or clock dependency, so identical inputs always return identical results.
This makes the calculator suitable both for quick manual checks and as a small deterministic building block inside a larger educational or algebraic workflow.
What you can do with it
Check a calculus exercise
Confirm the coefficient and exponent obtained after differentiating a single power term by hand.
Generate worked examples
Create dependable derivative pairs for worksheets, lesson notes, quizzes, or answer keys.
FAQ
Can the exponent be negative or fractional?
Yes. Any finite numeric exponent is accepted, and the same power-rule arithmetic is applied.
What happens when the exponent is zero?
The derivative coefficient is zero. The returned mechanical exponent is −1, but a zero coefficient means the derivative itself is zero.
Can I enter a polynomial with several terms?
No. This calculator accepts one coefficient and one exponent for a single term. Differentiate each polynomial term separately.
What is the derivative of 5x^4?
It is 20x^3. Multiply the coefficient 5 by the exponent 4 to get 20, then subtract one from the exponent to get 3. Enter 5 and 4 and the calculator returns exactly that pair.
Can it give the value of the derivative at a specific x?
No. It returns only the coefficient and exponent of the derivative of one term. It does not evaluate at a point, combine like terms or apply the product, quotient or chain rules.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/calculus/derivative-power-rule \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"coefficient":5,"exponent":4}'const res = await fetch("https://api.kit.forhosting.com/calculus/derivative-power-rule", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"coefficient": 5,
"exponent": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/derivative-power-rule",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"coefficient": 5,
"exponent": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/derivative-power-rule", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"coefficient":5,"exponent":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"coefficient":5,"exponent":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/derivative-power-rule", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"coefficient": 5,
"exponent": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.derivative_power_rule",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Enter the two parts of the power term
Separating the coefficient from the exponent makes the contract unambiguous for browser use and API automation. It also avoids assumptions about multiplication signs, Unicode superscripts, variable names, or whitespace that a symbolic expression parser would otherwise need to resolve.
Read the derivative coefficient and exponent
First, it multiplies the original coefficient by the original exponent, producing derivative_coefficient = a·n. Second, it subtracts one from the original exponent, producing derivative_exponent = n−1.
Know the scope and numerical limits
If multiplying a by n would produce infinity, or subtracting one from n would leave the finite range, the request is rejected instead of returning misleading JSON.
Integration use cases
Automate monomial differentiation
Send numeric coefficient-exponent pairs through the API and reuse the structured results in a larger workflow.
Technical questions
What formula does the calculator use?
It uses d/dx(a·x^n) = (a·n)·x^(n−1).
What does it cost to use through the API?
Each API request costs $0.002. The browser calculator is available for direct interactive use.