Polynomial Derivative Calculator

Differentiate polynomials with clear steps and accuracy. Check higher orders, slopes, and tangent equations instantly. Graph both curves for faster insight and confident verification.

Polynomial Derivative Calculator Form

Use the responsive fields below. Large screens show three columns, smaller screens show two, and mobile displays one.

Polynomial Degree

Choose the highest exponent present in the polynomial.

Derivative Order

Find first through eighth derivative values.

Evaluate at x

Used for function value, derivative value, and tangent line.

Coefficient of x^8

Coefficient of x^7

Coefficient of x^6

Coefficient of x^5

Coefficient of x^4

Coefficient of x^3

Coefficient of x^2

Coefficient of x^1

Coefficient of x^0

Graph Range Start

Graph Range End

Graph Step

Example Data Table

Polynomial	Order	x	Derivative Result	Value at x
3x^4 - 2x^3 + 5x - 7	1	2	12x^3 - 6x^2 + 5	77
3x^4 - 2x^3 + 5x - 7	2	2	36x^2 - 12x	120
-4x^5 + x^2 + 9	3	1	-240x^2	-240

Formula Used

General polynomial: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

Single-term derivative rule: d/dx [axⁿ] = a·n·x^n-1

kth derivative rule: d^k/dx^k [axⁿ] = a·n·(n-1)·...·(n-k+1)·x^n-k, for k ≤ n

Tangent line: y - f(x₀) = f'(x₀)(x - x₀)

How to Use This Calculator

Select the polynomial degree that matches your highest exponent.
Enter coefficients for each power of x, including zero values where needed.
Choose the derivative order you want to calculate.
Provide the x-value for function evaluation and derivative evaluation.
Set graph start, end, and step values for the plotted curves.
Press Calculate Derivative to view results above the form.
Review the derivative steps, preview table, and Plotly graph.
Use the CSV and PDF buttons to export the current result set.

Frequently Asked Questions

1) What does this calculator differentiate?

It differentiates polynomials with degree one through eight. You can enter any real-number coefficients, including decimals, negatives, and zeros.

2) Can it find higher-order derivatives?

Yes. You can calculate first through eighth derivatives. If the derivative order exceeds the polynomial degree, the result becomes zero.

3) Does it evaluate the derivative at a point?

Yes. Enter an x-value to compute the original function value, the first derivative value, and the selected derivative order value at that point.

4) Why is the tangent line based on the first derivative?

The tangent slope comes from f'(x). Higher-order derivatives describe curvature changes, but the tangent line itself always uses the first derivative.

5) What if some coefficients are zero?

That is completely fine. Zero coefficients simply remove those terms from the polynomial, while keeping the degree structure easy to enter.

6) Why does my derivative become zero?

Repeated differentiation lowers exponents. Once every remaining exponent falls below the selected derivative order, all terms vanish and the derivative is zero.

7) What does the graph show?

The graph compares the original polynomial, the selected derivative order, and the tangent line. This helps you inspect slope behavior and curve changes quickly.

8) What is included in the exports?

The export files include the main results and a preview of generated x-values with corresponding function and derivative outputs for quick sharing.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.