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Free calculus tool

Derivative Calculator

Type a function, see it typeset as you go, and take the derivative with the steps.

Derivative calculator

Original function

Use ^ for powers, and sqrt( ), sin( ), ln( ), e^x and pi. Multiplication signs are optional: 2x sin(x) works.

x2sin⁡(x)\displaystyle x^{2}\sin\left(x\right)
Which derivative

Try one

What you can type

The input reads ordinary calculator notation. Multiplication signs are optional, brackets can be round, square or curly, and every function below accepts its argument with or without parentheses.

Accepted notation, the spellings that work, and how each one renders.
NotationType it asRenders as
Powersx^3x**3x³x3\displaystyle x^{3}
Implicit multiplication2x sin(x)2*x*sin(x)2xsin⁡(x)\displaystyle 2x\sin\left(x\right)
Division(x + 1)/(x - 1)x+1x−1\displaystyle \frac{x + 1}{x - 1}
Rootssqrt(x)cbrt(x)x^(1/2)x\displaystyle \sqrt{x}
Exponentialse^(2x)exp(2x)2^xe2x\displaystyle e^{2x}
Natural logarithmln(x)ln⁡(x)\displaystyle \ln\left(x\right)
Other logarithmslog(x)log_2(x)log(2, x)log⁡(x)\displaystyle \log\left(x\right)
Trigonometrysin(x)cos 2xtan(x)sec(x)csc(x)cot(x)sin⁡(x)\displaystyle \sin\left(x\right)
Powers of functionssin^2(x)sin(x)^2sin⁡2(x)\displaystyle \sin^{2}\left(x\right)
Inverse trigonometryarcsin(x)asin(x)sin^-1 xarcsin⁡(x)\displaystyle \arcsin\left(x\right)
Hyperbolic functionssinh(x)cosh(x)tanh(x)sinh⁡(x)\displaystyle \sinh\left(x\right)
Absolute value|x - 1|abs(x - 1)∣x−1∣\displaystyle \left|x - 1\right|
Constantspieπ\displaystyle \pi
Greek variablestheta^2sin(alpha)θ2\displaystyle \theta^{2}

log means base 10 on this page, and ln is the natural logarithm. Use log_2(x) or log(2, x) for another base. WeBWorK treats log as the natural logarithm, so the plain-text answer below the result never uses log: it writes ln, or a ratio of ln, which every system reads the same way.

Any single letter is a variable, and so is a spelled-out Greek name like theta. When an expression contains more than one, choose which to differentiate with respect to; the rest are treated as constants.

The rules it applies

There is no guessing behind the answer. Each step below names the rule it used, and these are the rules it has:

Power rule

ddx[xn]=nxn−1\displaystyle \frac{d}{dx}\left[x^{n}\right] = n x^{n-1}

Any constant exponent, positive, negative or fractional. This is what turns 1/x into -1/x² and √x into 1/(2√x).

Constant multiple

ddx[c f(x)]=c f′(x)\displaystyle \frac{d}{dx}\left[c\,f(x)\right] = c\,f'(x)

A constant factor comes along for the ride. Letters other than the one you are differentiating count as constants.

Sum rule

ddx[f(x)±g(x)]=f′(x)±g′(x)\displaystyle \frac{d}{dx}\left[f(x) \pm g(x)\right] = f'(x) \pm g'(x)

Term by term, which is why a polynomial collapses one power at a time.

Product rule

ddx[uv]=u′v+uv′\displaystyle \frac{d}{dx}\left[uv\right] = u'v + uv'

Extended to as many factors as you type: each one takes its turn being differentiated.

Quotient rule

ddx[uv]=u′v−uv′v2\displaystyle \frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^{2}}

Applied whenever both the top and the bottom contain the variable, so the answer comes out as one fraction.

Chain rule

ddx[f(g(x))]=f′(g(x)) g′(x)\displaystyle \frac{d}{dx}\left[f(g(x))\right] = f'(g(x))\,g'(x)

The inner derivative that appears whenever something other than the bare variable sits inside a function.

When the variable appears in both the base and the exponent, as in x^x, neither the power rule nor the exponential rule applies on its own. The calculator uses logarithmic differentiation there, and the steps say so.

Elementary derivatives

And the elementary derivatives it composes with them. Each also picks up a chain-rule factor when its argument is not just the variable:

  • ddx[sin⁡x]=cos⁡x\displaystyle \frac{d}{dx}\left[\sin x\right] = \cos x
  • ddx[cos⁡x]=−sin⁡x\displaystyle \frac{d}{dx}\left[\cos x\right] = -\sin x
  • ddx[tan⁡x]=sec⁡2x\displaystyle \frac{d}{dx}\left[\tan x\right] = \sec^{2} x
  • ddx[sec⁡x]=sec⁡xtan⁡x\displaystyle \frac{d}{dx}\left[\sec x\right] = \sec x \tan x
  • ddx[cot⁡x]=−csc⁡2x\displaystyle \frac{d}{dx}\left[\cot x\right] = -\csc^{2} x
  • ddx[ex]=ex\displaystyle \frac{d}{dx}\left[e^{x}\right] = e^{x}
  • ddx[ax]=axln⁡a\displaystyle \frac{d}{dx}\left[a^{x}\right] = a^{x}\ln a
  • ddx[ln⁡x]=1x\displaystyle \frac{d}{dx}\left[\ln x\right] = \frac{1}{x}
  • ddx[arcsin⁡x]=11−x2\displaystyle \frac{d}{dx}\left[\arcsin x\right] = \frac{1}{\sqrt{1 - x^{2}}}
  • ddx[arccos⁡x]=−11−x2\displaystyle \frac{d}{dx}\left[\arccos x\right] = -\frac{1}{\sqrt{1 - x^{2}}}
  • ddx[arctan⁡x]=11+x2\displaystyle \frac{d}{dx}\left[\arctan x\right] = \frac{1}{1 + x^{2}}
  • ddx[∣x∣]=x∣x∣\displaystyle \frac{d}{dx}\left[\left|x\right|\right] = \frac{x}{\left|x\right|}

Questions

Does it show the steps?

Yes. Open "Show the steps" under any result and you get one line per rule, with the rule named and the expression as it stands after that rule is applied. The last line is the simplification. For a second or third derivative the steps are grouped by pass, so you can see the result of each one before the next begins.

What does log mean here?

log is base 10 and ln is the natural logarithm, which is how Stewart, Desmos and a graphing calculator read them. For another base type log_2(x) or log(2, x). WeBWorK is the exception: it treats log as the natural logarithm, so the plain-text answer this page gives you never uses log, only ln.

Why does my answer look different from the calculator's?

Two correct derivatives can be written many ways. This page keeps the factored shape the rules produce rather than multiplying everything out, so a quotient stays one fraction and a product stays a product. It also leaves trigonometric identities alone: sec squared x is not rewritten as 1 plus tan squared x. If you are not sure two forms agree, try both at a few values of x.

Can I use the answer on WeBWorK?

The plain-text line under each result is written in the syntax WeBWorK accepts, and the Copy button puts it on your clipboard. Use it to check your own work. Copying an answer you have not worked through is how a midterm goes badly, and it is the part a tutor can actually help with.

Is anything I type stored or sent anywhere?

No. The parser and the differentiation rules are JavaScript running in your browser, with no request to any server. Nothing you type is stored, logged or sent, and the page works with your connection off once it has loaded.

What can it not do?

It differentiates one variable at a time, treating other letters as constants, so it does not do partial derivatives, implicit differentiation or related rates. It has no integrals, limits or equation solving. Piecewise functions, summations and unknown functions like f(x) are not supported either.

Stuck on the why, not the answer?

Getting the derivative is the easy half. Knowing which rule a problem wants, and why your version came out different, is what midterms actually test. We tutor UBC and SFU first-year calculus and BC Calculus 12 one-on-one, and we start from the problem in front of you.

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