Derivative Calculator

Step-by-step derivative calculator.

Enter any function and get the derivative worked out rule by rule. See exactly how the product, power, chain, and other rules are applied — just like writing it out by hand.

Input expression

skip the f(x) =
your function, typeset
Variable
Order
Show steps
Simplify
Examples
Practice problem

Steps

each rule in the order it was applied

    Answer

    Graph

    Drag to pan, scroll or pinch to zoom, and click the graph to move the tangent point. Both curves are sampled in the browser; poles are left as gaps rather than joined.

    Zoom

    How Derivative Calculator works

    1. Parse

      Your text becomes a tree of operations, with implicit multiplication signs inserted where you left them out.

    2. Typeset

      The tree is rendered as notation so you can check the calculator read the same thing you meant.

    3. Differentiate

      One rule per node — sum, product, quotient, power, chain — each recorded as a step in the working.

    4. Simplify and plot

      The result is reduced with exact arithmetic, then both curves are sampled and drawn.

    Notation and rules

    Everything the parser accepts, and the rules the engine applies. Each rule name links to its own page.

    Rules applied

    Rule Statement
    Constant \(\frac{d}{dx}c = 0\)
    Variable \(\frac{d}{dx}x = 1\)
    Sum \(\left(f + g\right)' = f' + g'\)
    Constant multiple \(\left(cf\right)' = c\,f'\)
    Power \(\frac{d}{dx}x^{n} = n x^{n-1}\)
    Product \(\left(fg\right)' = f'g + fg'\)
    Quotient \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}\)
    Chain \(\left(f \circ g\right)' = f'(g)\,g'\)
    Exponential \(\frac{d}{dx}a^{x} = a^{x}\ln a\)
    Logarithm \(\frac{d}{dx}\ln x = \frac{1}{x}\)
    Logarithm, base a \(\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}\)
    Trigonometric \(\frac{d}{dx}\sin x = \cos x\)
    Inverse trig \(\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\)
    Hyperbolic \(\frac{d}{dx}\tanh x = \operatorname{sech}^{2} x\)
    Inverse hyperbolic \(\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^{2}}\)
    Absolute value \(\frac{d}{dx}\left|x\right| = \operatorname{sgn} x, \; x \neq 0\)
    Logarithmic \(\frac{d}{dx}f^{g} = f^{g}\left(g'\ln f + \frac{gf'}{f}\right)\)
    Rewrite in natural logs \(\log_{a} b = \frac{\ln b}{\ln a}\)

    Input syntax

    You type It means
    2x, 3(x+1) Implicit multiplication
    x^3, x**3 Powers — x² and x³ also work
    sqrt(x), √x Square root
    cbrt(x) Cube root
    e^x, exp(x) Exponential function
    ln(x), log(x) Natural logarithm
    log(2, x), log2(x) Logarithm to a given base
    sin^2(x) The square of sin x
    tan^-1(x), arctan(x) Inverse tangent
    |x-1|, abs(x-1) Absolute value
    pi, e, tau Constants
    x_1, theta Subscripted and Greek variables

    Questions

    Does the calculator show every step?

    Yes. Every rule the engine applies is recorded and listed in order, with the general law it used and a short explanation of why it applies. Long expressions are written out a batch at a time so the page stays quick.

    Which functions are supported?

    Polynomials and roots, exponentials and logarithms to any base, all six trigonometric functions and their inverses, the hyperbolic family and their inverses, plus the absolute value and sign functions.

    Can it take higher derivatives?

    Yes, up to the tenth. Each round is differentiated in full and the working for the first round is shown step by step.

    Is my input sent to a server?

    No. Parsing, differentiation, simplification and plotting all happen in your browser. Your recent expressions are kept in local storage on your own device.

    Can I use it on a phone?

    Yes. The layout stacks to a single column, the graph accepts touch gestures for panning and pinch zoom, and the function keyboard covers the symbols a phone keyboard makes awkward.