Quick answer
The derivative f′(x) gives the rate of change of f. For f(x) = x³ sin x the product rule gives 3x² sin x + x³ cos x, which is 5.4729 at x = 1.3.
Enter a function of x, the order and optionally a point. The calculator returns the derivative and its value at the point.
Related tools: Integral Calculator: Antiderivative of a function of x, with optional definite bounds. Limit Calculator: Two-sided limit of a function of x, with a numeric check. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Derivative Calculator
- Type function of x, for example x^3*sin(x).
- Enter order (a whole number, between 1 and 3). It starts at 1.
- Optionally, enter evaluate at x = (optional).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Function of x | Text, such as x^3*sin(x) | — |
| Order | A whole number | ≥ 1, ≤ 3, starts at 1 |
| Evaluate at x = (optional) (optional) | A number | — |
Formula and method
Formula
d/dx f(x), repeated for higher orders
Symbolic result; numeric values to six decimals.
Each step is verified against a numerical central-difference derivative of the previous expression before it is shown.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Function of x = x^3*sin(x); Order = 1; Evaluate at x = (optional) = 1.3.
| Result | Value |
|---|---|
| Derivative | 3*x^2*sin(x)+x^3*cos(x) |
| Value at x | 5.472935 |
Example 2
Inputs: Function of x = e^(2*x)/x; Order = 1; Evaluate at x = (optional) = 0.7.
| Result | Value |
|---|---|
| Derivative | 2*x^-1*e^(2*x)-x^-2*e^(2*x) |
| Value at x | 3.310367 |
Example 3
Inputs: Function of x = ln(x)*x^2; Order = 2; Evaluate at x = (optional) = 2.1.
| Result | Value |
|---|---|
| Derivative | 3+2*ln(x) |
| Value at x | 4.483875 |
The first line is the headline result (derivative). Below it the derivative Calculator lists value at x, which describe the same case from other angles. Symbolic result; numeric values to six decimals.
Rules applied
The product rule (uv)′ = u′v + uv′ gives 3x² · sin x + x³ · cos x. The calculator also uses the chain and quotient rules and the derivatives of powers, exponentials, logarithms and trigonometric functions.
Scope
It handles elementary functions of one variable x; anything the engine cannot simplify or differentiate is rejected rather than guessed. Higher orders repeat the process. Check results against a numerical estimate such as (f(x + h) − f(x))/h.
How the result changes with evaluate at x =
Holding every other input at the first worked example, this table changes only evaluate at x =. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Evaluate at x = | Derivative | Value at x |
|---|---|---|
| 0.65 | 3*x^2*sin(x)+x^3*cos(x) | 0.985698 |
| 1.3 | 3*x^2*sin(x)+x^3*cos(x) | 5.472935 |
| 2.6 | 3*x^2*sin(x)+x^3*cos(x) | -4.606309 |
| 5.2 | 3*x^2*sin(x)+x^3*cos(x) | -5.78865 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Function of x: left empty shows "Enter function of x." beside that field and no result.
- Order: entering "4" shows "Order must be at most 3." beside that field and no result.
Choosing between the Derivative Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Integral Calculator](/math/integral-calculator/) | function of x | Antiderivative |
| [Limit Calculator](/math/limit-calculator/) | function of x and x approaches | Limit |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
Background: calculus, vectors and matrices
A derivative measures how fast a function's output changes with its input, and an integral accumulates a quantity such as area. The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc, so [[3, 8], [4, 6]] has determinant 3×6 − 8×4 = −14.
The dot product of two vectors is the sum of the products of matching components and is a single number; a positive value means the vectors point in broadly the same direction.
Assumptions and limitations
- Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for function of x and order to confirm this.
Frequently asked questions
What is the derivative of x²?
The derivative of x² is 2x; at x = 3 the slope is 6.
What is the second derivative?
The derivative of the derivative, a measure of curvature.
Can I use e^x or ln x?
Yes. e^x differentiates to itself and ln x to 1/x, and the calculator accepts both in an expression.
What does the Derivative Calculator calculate?
Derivative of a function of x, with a numeric check. It returns Derivative and Value at x.
What formula does the Derivative Calculator use?
d/dx f(x), repeated for higher orders. Each step is verified against a numerical central-difference derivative of the previous expression before it is shown.
What do I need to enter in the derivative Calculator?
You enter Function of x and Order. Optional: Evaluate at x = (optional).
How is the result rounded or limited?
Symbolic result; numeric values to six decimals. Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported.
What is an example of a derivative result?
With function of x x^3*sin(x), order 1 and evaluate at x = (optional) 1.3, the derivative is 3*x^2*sin(x)+x^3*cos(x).
Where does the method come from?
The method follows: Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy.
Related calculators
- Integral Calculator — Antiderivative of a function of x, with optional definite bounds.
- Limit Calculator — Two-sided limit of a function of x, with a numeric check.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Area Calculator — Area of common two-dimensional shapes.
Sources
- Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.