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What is a derivative?What is an integral?Derivative rulesIntegration rulesThe power rule

Derivative Calculator

Type a function and get its derivative with every rule named, then drag the tangent line along the curve to see what the answer means.

ddx\frac{d}{dx}
Reads as x3−3x2+2x^{3} - 3 x^{2} + 2
Try:
-2-11234-40-30-20-1010203040
  • f(x)

f(1.5) = -1.375   slope f′(1.5) = -2.25

What the derivative tells you

The derivative f′(x)f'(x) is the slope of the curve at each point: how fast ff is changing right there. On the graph above, the dashed amber curve is f′f'. Wherever the pink tangent line is flat, the amber curve crosses zero; where the tangent is steep, the amber curve is far from zero. Formally it is the limit of the slope of secant lines, f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}, and you can watch that limit happen on what is a derivative?

How the calculator works it out

CalcViz reads your expression into a tree (sums, products, powers and functions) and walks that tree applying one rule per node, which is exactly how you would do it on paper:

  • Sum rule: split (u+v)′(u+v)' into u′+v′u' + v'.
  • Constant multiple: (c u)′=c u′(c\,u)' = c\,u'.
  • Power rule: (xn)′=nxn−1(x^n)' = n x^{n-1} for any real nn, including roots and negative powers. See the power rule.
  • Product and quotient rules: (uv)′=u′v+uv′(uv)' = u'v + uv' and (uv)′=u′v−uv′v2\left(\tfrac{u}{v}\right)' = \tfrac{u'v - uv'}{v^2}.
  • Chain rule: for a function inside a function, multiply the outer derivative by the inner one. The chain rule calculator shows the split explicitly.

The full list with visual proofs is on derivative rules. For a quick answer on a single function, see the reference pages, e.g. the derivative of ln x or the derivative of sin x.

Worked example

Differentiate f(x)=x2sin⁡xf(x) = x^2\sin x. It is a product, so take u=x2u = x^2 and v=sin⁡xv = \sin x. Then u′=2xu' = 2x and v′=cos⁡xv' = \cos x, so f′(x)=2xsin⁡x+x2cos⁡xf'(x) = 2x\sin x + x^2\cos x. Click the example chip above to see the same steps generated automatically.

Questions students ask

How do I type my function?

Use ^ for powers (x^3), * or a space for multiplication (3x and 3*x both work), sqrt() for roots, and sin, cos, tan, ln, log, exp, arctan and abs as functions. You can write sin^2(x) for (sin x)². Implicit multiplication like 2x sin(x) works.

Which differentiation rules does it show?

The power, constant, constant-multiple, sum, product, quotient and chain rules, plus the rules for exponentials, logarithms, trig, inverse trig and hyperbolic functions. Each step names the rule it used.

What does the pink line on the graph mean?

It is the tangent line at the point you choose. Its slope is the value of the derivative there, so the derivative is literally "the slope of the ruler laid against the curve". Drag across the graph to move it.

Can it find second and third derivatives?

Yes. Pick the order (1st, 2nd or 3rd). For second derivatives and concavity in more detail, use the second derivative calculator.

Why does my answer look different from the textbook’s?

Equivalent expressions can be written many ways: 1x2\frac{1}{x^2} and x−2x^{-2} are the same function. Check by plugging in a number, or look at the graph: equal functions draw the same curve.

Is it free, and is my work stored?

Free, no account. All maths runs in your browser. Your recent problems are kept only in this browser’s local storage, and you can clear them any time.