This tool differentiates a polynomial one term at a time and then builds the tangent and normal at a point you choose. It suits students who can follow the power rule in class but want to check their own working and see what the derivative means on a graph.
Free tool · Additional Mathematics
Polynomial differentiation tutor
You know the power rule but the tangent and normal steps still feel slippery.
Everything you enter stays on this device.
How do I use it?
- Type the coefficients from the highest power to the constant, separated by commas.
- Type the x value where you want the tangent.
- Press “Differentiate”.
- Read the term-by-term steps, then the tangent, the normal, the stationary points and the plot.
How does it work?
Each term ax^n becomes n × a × x^(n−1). A constant becomes 0. The tool writes that step for every term, then joins the results into f′(x).
For the tangent at x = a, it finds the point f(a) and the gradient f′(a). The tangent is y − f(a) = f′(a)(x − a). The normal uses the gradient −1 ÷ f′(a), and the tool reports a vertical line if f′(a) is 0.
When f′(x) is linear or quadratic, it solves f′(x) = 0 and uses the sign of f″(x) to label each stationary point as a minimum or a maximum.
A short worked example
Type 3, 2, 1 for f(x) = 3x² + 2x + 1, with the tangent at x = 1.
The power rule gives 3x² → 6x, 2x → 2 and 1 → 0, so f′(x) = 6x + 2. At x = 1, the point is (1, 6) and the gradient is 8.
The tangent is y − 6 = 8(x − 1), which simplifies to y = 8x − 2. The normal has gradient −1/8.
What are its limits?
The tool handles polynomials up to degree 6 only and has no expression parser, so it cannot read brackets, roots or fractions in x. It does not lay out answers the way an exam expects. The plot is a sketch of the idea, not an exact graph.
Where does it connect to the lessons?
Use it alongside differentiating powers, products and quotients, then move on to finding tangents and normals. Test yourself on the differentiation practice page.
If algebra slips are behind your errors, read should I repair algebra before revising differentiation. For a teacher to work with you, see SPM Additional Mathematics tuition and the one-hour trial class (from RM50).
Common questions
How do I type 2x³ − 5x + 4?
Enter the coefficients from the highest power down, with 0 for any missing power: 2, 0, -5, 4. The tool then rewrites your polynomial so you can check it was read correctly.
Why does a constant differentiate to zero?
A constant has a flat graph, so its gradient is 0 everywhere. Using the power rule, a constant is a × x⁰, and multiplying by the power 0 gives 0.
What is the normal?
The normal is the line at right angles to the tangent at the same point. Its gradient is −1 divided by the tangent gradient, and when the tangent is horizontal the normal is a vertical line.
Can it differentiate products, quotients or chain rule questions?
Not directly. Expand a product into a polynomial first if you want to use the tool. Those methods have their own lessons on this site.
A teacher working one-to-one can check whether the trouble is the power rule itself or the algebra around it, and start from the step that is actually shaky.
- Online one-to-one lessons for your child with an experienced teacher.
- Your first class is a one-hour trial, from RM50. The fee is agreed before you book.
- Happy with the teacher? Continue with lessons of about 1.5 hours. If not, ask for another teacher.