Derivative Calculator with Steps
The exact derivative of a function, term by term, naming the rule used at each step.
Results
What this tool does
Differentiating is mechanical: there is a rule for every shape an expression can take, and applying them in the right order always works. What takes practice is seeing which rule applies where. This page reads your function into a tree, applies the rules exactly as a textbook states them — sum, product, quotient, chain, power — and shows you which one it used on each term. The answer is exact symbolic algebra, not a numerical estimate, which is why it handles things like x^x that no approximation could get right.
Formula
(u ± v)′ = u′ ± v′ · (u v)′ = u′v + u v′ · (u ÷ v)′ = (u′v − u v′) ÷ v² · f(u)′ = f′(u) u′ · (uⁿ)′ = n uⁿ⁻¹ u′
Variables
| Symbol | Meaning | Unit |
|---|---|---|
expr | The expression, in x | — |
order | Which derivative | — |
at | Value it at x = | — |
DV | The derivative | — |
ST | Term · rule used · its derivative | — |
D1 | First derivative | — |
D2 | Second derivative | — |
VD | The derivative at that point | — |
VF | The function at that point | — |
V2 | The second derivative there | — |
TG | Tangent line at that point | — |
IN | How the page read it | — |
NT | Terms | — |
Worked example
- The expression, in x3x^2 + sin(2x) - ln(x)
- Which derivative1
- Value it at x =1
- The derivative6x + 2cos(2x) - 1 / x
- Term · rule used · its derivative 3x^2 (c*u)' = c*u' 6x sin(2x) (sin u)' = cos(u)*u' 2cos(2x) -ln(x) (ln u)' = u' / u -1 / x
- First derivative6x + 2cos(2x) - 1 / x
- Second derivative6 - 4sin(2x) + 1 / x^2
- The derivative at that point4.1677063269
- The function at that point3.9092974268
- The second derivative there3.3628102927
- Tangent line at that pointy = 4.1677063269x - 0.2584089001
- How the page read it3x^2 + sin(2x) - ln(x)
- Terms3
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
Frequently asked questions
How do I write the function?
The way you would on paper, with a few conventions. Use x for the variable and ^ for powers. Multiplication can be left out where it is obvious: 3x, 2sin(x) and x(x+1) are all understood. The functions available are sin, cos, tan, exp, ln, log, sqrt, sinh, cosh and tanh, and each needs its brackets: sin(2x), not sin 2x. The constants pi and e are recognised. If the page says it does not know a letter, that letter is not x and not one of those function names — a common cause is writing a problem in t on a page that works in x.
Is this an approximation?
No. The page reads your expression into a tree and applies the differentiation rules to that tree — the same rules you would apply by hand, in the same order. The result is exact symbolic algebra, not a numerical estimate taken from nearby points. That is why it can differentiate x^x correctly, which a numerical method would only approximate. The only place floating point enters is when you ask for the value at a particular point, and there it is ordinary arithmetic.
Why does the answer not look factorised?
Because the page simplifies but does not factorise, and the distinction matters. It will collect a polynomial into its standard form, cancel zeros and ones, fold arithmetic between numbers and write awkward decimals as fractions — so x^6/6 rather than 0.1666666667x^6. What it will not do is spot that 2x·exp(x) + x²·exp(x) can be written x·exp(x)(2 + x), because deciding which factorisation is the nicer one is a judgement rather than a rule. The form you see is what the product rule actually produces, which is also what a marker expects to see before any tidying.