How to use it
- Type the function, for example x^2*sin(x).
- Choose the first, second or third derivative.
- Click Differentiate.
- Read the result and the steps; copy it if you need it.
What the calculator does
It differentiates symbolically, simplifies the result, and then checks it numerically: the derivative is compared with a difference quotient of the original function at several points, and a badge shows whether it passed. For the first derivative it also lists the steps: which rule applies to each part of the function and what the parts' derivatives are.
Example: f(x) = x² sin(x). The product rule (u·v)′ = u′·v + u·v′ with u = x² and v = sin(x) gives
f′(x) = 2x·sin(x) + x²·cos(x).
For the second and third derivatives, the same rules are applied to the previous result.
Differentiation rules used
| Rule | Formula |
|---|---|
| Sum and difference | (u ± v)′ = u′ ± v′ |
| Constant multiple | (c·u)′ = c·u′ |
| Power rule | (xⁿ)′ = n·xⁿ⁻¹ |
| Product rule | (u·v)′ = u′·v + u·v′ |
| Quotient rule | (u / v)′ = (u′·v − u·v′) / v² |
| Chain rule | (f(u))′ = f′(u)·u′ |
| Exponential | (eᵘ)′ = eᵘ·u′, (aᵘ)′ = aᵘ·ln a·u′ |
| Logarithmic differentiation | (uᵛ)′ = uᵛ·(v′·ln u + v·u′/u) |
Basic derivatives: (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = 1/cos² x, (ln x)′ = 1/x, (eˣ)′ = eˣ, (√x)′ = 1/(2√x), (arctan x)′ = 1/(1 + x²).
How to enter functions
Type functions the way you'd write them: x^3, sqrt(x) or √x, 2x or 3*x, sin(x)^2 or sin^2(x), e^x or exp(x), ln(x) for the natural log, log10(x), abs(x), and the constants pi and e. Implicit multiplication like 2x, x(x+1) and sinx is understood.
If the input can't be parsed, the calculator tells you where; check parentheses and operators. Click an example below the input box to try it.
Worked examples
- (x³ − 2x + 1)′ = 3x² − 2 (power rule term by term).
- (sin(x²))′ = 2x·cos(x²) (chain rule with u = x²).
- (ln(x) / x)′ = (1 − ln x) / x² (quotient rule).
- (e^(2x)·cos x)′ = e^(2x)·(2cos x − sin x) (product and chain rule).
- (xˣ)′ = xˣ·(ln x + 1) (logarithmic differentiation).
The results may be arranged differently from a textbook, since there are many equivalent forms; the numerical check confirms they're the same function.
Uses and limits
Use it to check homework, verify a step in a longer problem, or get higher derivatives for Taylor series and curve sketching (f″ for concavity and inflection points). Very unusual functions may not have a symbolic form the engine can produce; in that case it says so instead of guessing. The math engine loads in your browser, and calculations run on your device.
Frequently asked questions
Does the derivative calculator show steps?
Yes. For the first derivative it lists each rule used (product, quotient, chain rule and so on) and the derivatives of the parts.
Can it find the second derivative?
Yes. Choose second or third derivative; each is computed from the previous result.
How do I type e^x and ln?
Type e^x (or exp(x)) and ln(x). Use log10(x) for the base-10 logarithm.
Is the answer checked?
Yes. The result is compared numerically with the original function's rate of change at several points.
Can I differentiate with respect to a variable other than x?
The calculator uses x as the variable; other letters are treated as unknown symbols, so write constants as numbers.
Last updated: November 12, 2026