How to use it
- Type the function in Function f(x), for example x^3 - 2x + 1. Write 2x or 2*x for two times x.
- Check that Variable holds the letter the function uses, usually x.
- Enter the Point where you want the slope.
- Read Derivative f′(x) for the general formula and Derivative at the point for its value. Open Working, rule by rule to see which rule gave each part.
How the derivative is found
The derivative of f at x is the rate at which f changes, which is the slope of its graph at x. The calculator differentiates the expression as you typed it, one rule at a time, simplifies the result, and then puts the point into it.
Each line of the working names the rule it uses. The rules are the standard ones: they are stated in an open calculus text, listed in the sources below.
Power rule: (x^n)′ = n·x^(n−1)
Sum rule: (f + g)′ = f′ + g′
Product rule: (f·g)′ = f′·g + f·g′
Quotient rule: (f/g)′ = (f′·g − f·g′) ÷ g²
Chain rule: (sin f)′ = cos(f)·f′Which functions the calculator reads
- Numbers, the variable (one letter), pi and e.
- Sums, differences, products, quotients and powers of these.
- The functions sin, cos, tan, exp (e to a power), ln (natural logarithm), log (base 10) and sqrt. Each one is written with its brackets.
- Implied products such as 2x or 3(x + 1). Write a space or * between a letter and a function name, such as x*sin(x).
What this calculator does not cover
- Functions of two variables, implicit functions and differential equations.
- Points where the function has no finite slope or is not defined. At those points no value is given.
- Functions not listed above, such as arcsin or sinh.
- Integrals. The definite-integral calculator on this site works out the area under a graph between two limits.
Worked example: f(x) = x³ − 2x + 1 at x = 2
The sum rule splits the function into x³ − 2x and 1. The power rule gives (x³)′ = 3x², the constant multiple rule gives (2x)′ = 2, and the constant rule gives (1)′ = 0.
The derivative is 3x² − 2. At x = 2 it is 3 · 2² − 2 = 12 − 2 = 10, so the graph of f rises with slope 10 at that point.
| Result | Value |
|---|---|
| Derivative f′(x) | 3x^2 - 2 |
| Derivative at x = 2 | 10 |
Frequently asked questions
What does the derivative tell me?
It is the slope of the graph of the function at each point: how fast the output changes when x changes by a tiny amount. A positive derivative means the function is rising at that point, and a negative one means it is falling.
Do I need to write 2*x, or is 2x enough?
Both work. The calculator reads 2x as two times x, as a textbook writes it. Write a space or * between a letter and a function name, such as x*sin(x), because xsin would be read as one unknown name.
Can I use a comma for the decimal point?
Yes. 2,5x and 2.5x mean the same thing. Do not use commas to separate numbers: the function is read as one expression.
Why does the calculator say there is no value at this point?
Some functions are not defined everywhere: ln needs a positive number, and a square root needs zero or more. Where the slope is infinite, such as the square root at zero, no finite value exists, so none is given.
Is the derivative the same as the integral?
No. The derivative finds the slope of a graph, and the integral adds up the area under it. The definite-integral calculator on this site works out that area between two limits.
Does the calculator send my function anywhere?
No. The function is read and differentiated in your browser. Nothing you type is stored or sent to us.
Sources
- Lehigh University, Math 21 online notes: power, product, quotient and chain rules
- OpenStax, Calculus Volume 2: Table of Derivatives (sin, cos, tan, e^x, ln, b^x, log base b)
Figures and text last checked on Thursday, November 12, 2026.