Enter coefficients from highest power to constant. Example: 3 2 1 means
3x² + 2x + 1. Fractions like 1/2 are also allowed.
Example function: f(x) = (3x² + 2x + 1) / (x² + 1)
| x | Numerator | Denominator | f(x) | f′(x) |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 2 |
| 1 | 6 | 2 | 3 | 1 |
| 2 | 17 | 5 | 3.4 | 0.08 |
| 3 | 34 | 10 | 3.4 | -0.04 |
f′(x) = [u′(x)v(x) − u(x)v′(x)] / [v(x)]²
- Differentiate the numerator polynomial to get u′(x).
- Differentiate the denominator polynomial to get v′(x).
- Multiply u′(x) by v(x).
- Multiply u(x) by v′(x).
- Subtract the second product from the first product.
- Square the denominator polynomial.
- Divide the new numerator by the squared denominator.
- Check the domain because the denominator cannot equal zero.
- Enter numerator coefficients in descending powers, such as
4 -3 2. - Enter denominator coefficients the same way, such as
1 0 -5. - Choose the x-value where you want the function and slope evaluated.
- Set the graph interval and point density for the Plotly chart.
- Pick the decimal precision for cleaner tables and displayed results.
- Press Calculate Quotient Derivative to view the answer above the form.
- Review the symbolic derivative, substituted values, tangent line, and computed sample table.
- Use the export buttons to download your result data as CSV or PDF.
1) What does the quotient rule calculate?
It finds the derivative of one function divided by another. This calculator applies the quotient rule to polynomial numerators and denominators, then evaluates the result at a chosen x-value.
2) Why are coefficients entered from highest power first?
Descending order lets the calculator correctly interpret polynomial structure. For example, 3 2 1 becomes 3x² + 2x + 1, which makes differentiation and evaluation consistent.
3) Can I enter fractions or negative coefficients?
Yes. You can use negative numbers and simple fractions like -3/2 or 1/4. This helps when your polynomial comes from exact algebraic work rather than decimal approximations.
4) Why does the result sometimes say undefined?
A quotient is undefined wherever the denominator equals zero. The derivative can also be undefined there because the squared denominator in the quotient rule becomes zero as well.
5) What does the Plotly graph show?
It plots the original quotient function, its derivative, and the tangent line at your selected x-value when that point exists. Undefined locations are intentionally broken in the curve.
6) Does the calculator simplify the derivative?
It expands the quotient-rule numerator and the squared denominator polynomial. That gives a clean rational derivative form, which is usually enough for checking work, plotting behavior, and numeric evaluation.
7) What is the tangent line used for?
The tangent line shows local linear behavior near the chosen x-value. It helps you visualize slope, estimate nearby values, and compare the function to its linear approximation.
8) Is this useful for study and homework checking?
Yes. It shows the symbolic rule, intermediate derivatives, substituted values, graph, and exportable table. That makes it useful for verification, revision, and step-by-step practice.