Input expression
skip the f(x) =Steps
each rule in the order it was appliedAnswer
Second and higher-order derivatives
Differentiate again to get acceleration, concavity, and the test that classifies turning points.
Differentiating twice gives the second derivative, written \(f''(x)\), \(\frac{d^{2}y}{dx^{2}}\) or \(y''\). It is the rate at which the rate is changing: if \(f\) is position and \(f'\) is velocity, then \(f''\) is acceleration. Set the Order control on the calculator above to 2 — or use the smaller one here — and the working shows every round.
What the second derivative tells you
- \(f'' > 0\): the curve is concave up, holding water, and \(f'\) is increasing.
- \(f'' < 0\): the curve is concave down, and \(f'\) is decreasing.
- \(f'' = 0\) with a sign change on either side: an inflection point, where the curvature flips.
The second derivative test
At a point where \(f'(a) = 0\), the second derivative classifies it without checking signs on both sides: \(f''(a) > 0\) means a minimum, \(f''(a) < 0\) means a maximum, and \(f''(a) = 0\) is inconclusive — fall back to the first derivative test there.
Third, fourth and nth derivatives
Keep going and the notation switches from primes to \(f^{(4)}\), \(f^{(5)}\), \(f^{(n)}\). Some functions settle into a pattern: \(e^{x}\) is unchanged at every order, \(\sin x\) cycles back to itself every four, and a polynomial of degree \(n\) reaches zero at order \(n+1\). Others grow without ever repeating — a fifth-order derivative of a product can run to a hundred steps, which is why the working is written out a batch at a time.
How Derivative Calculator works
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Parse
Your text becomes a tree of operations, with implicit multiplication signs inserted where you left them out.
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Typeset
The tree is rendered as notation so you can check the calculator read the same thing you meant.
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Differentiate
One rule per node — sum, product, quotient, power, chain — each recorded as a step in the working.
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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 |
More derivative tools
The calculator above handles ordinary derivatives to any order. These pages take the same engine into the cases that need their own controls or their own explanation.
Partial derivatives
Work with two or three variables and differentiate with respect to any one of them.
Derivative at a point
Get the slope as a number at any x, and the equation of the tangent line there.
Derivative graphs
Plot a function against its derivative and read the turning points straight off.
Implicit differentiation
Find dy/dx for curves such as x² + y² = 25 that are not solved for y.
Parametric and polar
Curves given by x(t), y(t) or by r(θ), where dy/dx comes from the parameter.
Directional derivatives
The gradient of a surface, and how steeply it rises in any direction you choose.
Numerical differentiation
Estimate a derivative from values alone, and compare it with the exact answer.
Derivative of an integral
The Fundamental Theorem, and the chain rule for a variable limit.
Learn the mathematics
Explanations rather than tools, for when the answer is not the point.
What is a derivative?
The definition, the notation, and where derivatives sit in each calculus course.
First principles
The limit definition worked slowly, for x², √x and sin x.
Practice problems
Problems grouped by rule, with answers and the standard derivatives table.
Formula cheat sheet
Every rule and standard derivative on one printable page.
Looking for a specific rule? Rules lists all eighteen, each with worked examples and a graph.
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.
How do I use this derivative calculator?
Type the function — skip the "f(x) =" part — and press Differentiate. Implicit multiplication is understood, so 2x and 3(x+1) work. Pick the variable and the order if you need something other than the first derivative in x.
Can it handle constants and unknown parameters?
Yes. Any symbol that is not the differentiation variable is treated as a constant, so a, k and c behave like numbers. Use the variable selector to choose which symbol you are differentiating with respect to.
Does it work offline?
Once the page has loaded, yes. The parser, the differentiator, the simplifier and the plotter are all JavaScript running on your own device, so no request is made when you press Differentiate.
Can I copy the answer as LaTeX?
Yes — Copy LaTeX puts the typeset source on your clipboard, and Copy plain text gives an ASCII form. Share copies a link that reopens the same expression.
How do I find where the derivative is zero?
Stationary points are marked on the graph automatically. Set the tangent point near one and the readout shows a slope of zero and reports the point as stationary.
Is a derivative Calculus 1 or Calculus 2?
Calculus 1. Derivatives are introduced and completed there for functions of one variable; Calculus 2 applies them, and Calculus 3 extends them to several variables.