Polynomial Derivative
Enter a polynomial in x using ^ for exponents. The calculator differentiates it exactly, applying the power rule d/dx(c·xⁿ) = c·n·xⁿ⁻¹ to each term and dropping constants. Every term is shown so you can follow the differentiation.
One rule, applied term by term
Differentiating a polynomial is the one case where calculus is purely mechanical. No rule has to be chosen, nothing has to be recognised, and the answer is another polynomial one degree lower. Multiply each coefficient by its exponent, reduce the exponent by one, discard the constant.
This calculator does exactly that, and shows each term's transformation on its own line so the pattern is visible rather than inferred from the finished answer.
A polynomial is a sum, and differentiation distributes across a sum, so each term can be handled in complete isolation from the others. That independence is what makes the operation mechanical: there is no product rule, no chain rule and no ordering to worry about.
Listing the terms separately makes a disagreement easy to localise. A wrong answer is one wrong term, and the line-by-line output identifies which one without any need to redo the rest.
How to use this calculator
- Type the polynomial in x Use ^ for exponents and combine terms with plus and minus signs. Coefficients may be negative or fractional, and terms need not be in any particular order.
- Check the parsed function line The polynomial is echoed as the parser understood it, with terms collected and ordered. Comparing it against what you typed catches a mistyped exponent before it propagates.
- Read the per-term lines Each non-zero term gets a line showing the power rule applied to it. The constant gets its own line stating that its derivative is zero.
- Read the assembled derivative The terms collected into a single polynomial, one degree lower than the input unless the input was constant.
The formula, and where it comes from
d/dx(c·xⁿ) = c·n·xⁿ⁻¹ d/dx(constant) = 0 (f + g)′ = f′ + g′
The power rule does two things in one motion, and both are needed: the coefficient is multiplied by the old exponent, and the exponent is then reduced by one. Performing only the visible half — lowering the power — is the classic incomplete application.
The constant term is the case where the rule produces nothing rather than something small. Its exponent is zero, so multiplying by it annihilates the coefficient, and the term disappears from the answer entirely. The tool reports that explicitly rather than silently omitting it.
Linearity is what licenses handling each term alone. The derivative of a sum is the sum of the derivatives, so no interaction between terms is possible and the order of the work is irrelevant.
Everything here is exact. The implementation works from the coefficient list rather than manipulating a string, so the arithmetic is a single multiplication per term and no approximation enters at any stage.
What each input means
- f(x) Polynomial — form field “Polynomial in x”
- The expression to differentiate, in x with non-negative whole exponents. It is parsed into a coefficient list before any work is done.
- c, n Coefficient and exponent
- For each term. The new coefficient is their product and the new exponent is one less than the old.
- f′(x) Derivative
- The assembled result, one degree lower than the input. Exact, not approximated.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A cubic with all four terms present
A polynomial with a cubic, a quadratic, a linear and a constant term. Every kind of term appears exactly once.
Inputs Polynomial in x = 3x^3 - 5x^2 + 2x - 7
- Function f(x) = 3x³ − 5x² + 2x − 7
- Constant term d/dx(-7) = 0
- Power rule d/dx(2x) = 2
- Power rule d/dx(-5x²) = -10x
- Power rule d/dx(3x³) = 9x²
- Derivative f'(x) = 9x² − 10x + 2
Result f'(x) = 9x² − 10x + 2
Four lines appear but only three contribute to the answer: the constant's line records that its derivative is zero. The result is a quadratic, one degree lower, which is always the case for a genuine cubic.
The linear term's exponent of one drops to zero, so it becomes a bare constant with no x attached. That is the term whose transformation is most often written incorrectly, since the variable vanishing looks like a mistake.
A quartic with a gap
A polynomial in even powers only, so the cubic and linear terms are absent. Missing terms are simply skipped.
Inputs Polynomial in x = x^4 - 2x^2 + 1
- Function f(x) = x⁴ − 2x² + 1
- Constant term d/dx(1) = 0
- Power rule d/dx(-2x²) = -4x
- Power rule d/dx(1x⁴) = 4x³
- Derivative f'(x) = 4x³ − 4x
Result f'(x) = 4x³ − 4x
Only two power-rule lines appear alongside the constant, because terms with a zero coefficient are not listed. The derivative is a cubic with a gap of its own, since differentiating even powers gives odd ones.
Each surviving exponent drops by exactly one, so the parity of every power flips. An even polynomial always differentiates to an odd one, which is a quick structural check on the answer.
Reading the result
The degree always falls by one
A polynomial of degree n differentiates to one of degree n − 1, unless it was constant to begin with, in which case the derivative is zero. Any answer of the same degree as the input contains an unreduced exponent.
The constant term is gone for good
Its disappearance is why integration needs an arbitrary constant afterwards: differentiation destroys the information, so nothing can recover it. Two polynomials differing only in their constant have identical derivatives.
What the derivative is for
It gives the slope of the original curve at every point. Setting it to zero locates the turning points, and its sign says where the polynomial is rising or falling.
When you would use this
Finding turning points
The stationary points of a polynomial are the roots of its derivative, so differentiating is the first step in any maximum or minimum question. The resulting polynomial is one degree easier to solve than the original.
Checking worked differentiation
The per-term breakdown means a disagreement can be traced to a single term rather than prompting a full redo, which is the fastest way to find a dropped factor.
Assumptions and limitations
What this calculator assumes
- The input is a polynomial in x with non-negative whole-number exponents.
- Differentiation is with respect to x, and the result is exact.
- Terms are handled independently, which linearity guarantees is valid.
- Terms with a zero coefficient are omitted from the per-term output.
Where it stops being the right tool
- Polynomials only: no trigonometric, exponential or logarithmic terms, and therefore no chain, product or quotient rule.
- No negative or fractional exponents, which would take the input outside the power rule as implemented here.
- First derivative in one pass — a second derivative means feeding the result back in.
- One variable, and no evaluation of the derivative at a point.
Common mistakes
Reducing the exponent without multiplying by it
Why it happens. Lowering the power is the visible half of the rule and the half that changes the shape of the term. The multiplication leaves the term looking similar, so skipping it is easy under time pressure.
How to avoid it. Compare one term against its line in the output. If your exponents are right but the coefficients are too small, the multiplication is what went missing.
Keeping the constant term
Why it happens. Every other term survives in some form, so carrying the constant along feels consistent. Its derivative is zero and it should not appear at all.
How to avoid it. Count the terms. A derivative has one fewer than the original whenever there was a non-zero constant.
Differentiating the linear term to zero
Why it happens. Its exponent drops to zero and the x disappears, which makes the term look as though it has vanished the way the constant did.
How to avoid it. The coefficient survives as a plain number. Only a term with no x at all differentiates away entirely.
Learn why this works
Key terms
Frequently asked questions
What is the power rule?
The derivative of c·xⁿ is c·n·xⁿ⁻¹ — multiply the coefficient by the exponent, then reduce the exponent by one. A constant term has exponent zero, so the multiplication annihilates it and its derivative is zero.
How do I type a polynomial?
Use ^ for exponents and join terms with plus and minus, for example 3x^3 - 5x^2 + 2x - 7. Terms may be given in any order, and the parsed line shows how the input was understood.
Is the result exact?
Yes. The calculation works from the coefficient list and applies one multiplication per term, so it is symbolic and exact rather than a numerical estimate. Nothing is sampled or approximated.
Can I get the second derivative?
Not in one step, but the output is itself a polynomial in the same format. Feeding it back in differentiates again, and repeating eventually reaches zero.