Grafolyn

Derivative Calculator

Differentiate any function and see which rule was used at each step, then integrate it, find its turning points, or take a limit — all from the same expression. Everything is plotted, everything exports, and nothing is sent to a server.

Try:

First to fourth derivatives, with the rule used at each step

d/dx [ x³ − 6x² + 9x + 1 ]

3x² − 12x + 9

f′(x) = 3*x^2 - 12*x + 9

f′(1) = 0 — the slope of the tangent to f at x = 1.

  1. Sum ruled/dx[f + g] = f' + g'applied to x³ − 6x² + 9x + 1d/dx[x³ − 6x² + 9x] + d/dx[1]
  2. Sum ruled/dx[f + g] = f' + g'applied to x³ − 6x² + 9xd/dx[x³ − 6x²] + d/dx[9x]
  3. Difference ruled/dx[f − g] = f' − g'applied to x³ − 6x²d/dx[x³] − d/dx[6x²]
  4. Power ruled/dx[xⁿ] = n·xⁿ⁻¹applied to x³3x²
  5. Constant multiple ruled/dx[c · f] = c · f'applied to 6x²6d/dx[x²]
  6. Power ruled/dx[xⁿ] = n·xⁿ⁻¹applied to x²2x
  7. Constant multiple ruled/dx[c · f] = c · f'applied to 9x9d/dx[x]
  8. Variabled/dx[x] = 1applied to x1
  9. Constant ruled/dx[c] = 0applied to 10

What this covers

Four panels over one expression. Type a function once and you can differentiate it, integrate it, analyse its shape, and take limits of it without retyping — which matters more than it sounds, because retyping a function is where a surprising share of wrong answers come from.

Derivatives, with the working

First through fourth derivatives, each one simplified the way a person would write it rather than the way an algorithm leaves it. Underneath the answer is the list of rules that produced it — power rule, product rule, quotient rule, chain rule — each shown in its general form, the sub-expression it was applied to, and what came out.

That list is the reason to use this rather than a search result. An answer you cannot reproduce is worth very little in an exam. Seeing that the chain rule was applied to sin(2x) with inner function 2x, producing 2cos(2x), is something you can do again on the next question without any help.

You can also evaluate the derivative at a point, which is the same thing as asking for the slope of the tangent there, and plot f, f′ and f″ together. Plotting them together is worth doing at least once: the relationship between a curve and its derivative — f′ crossing zero exactly where f turns, f′ positive exactly where f rises — stops being a rule to memorise the moment you have seen it.

Integrals, symbolic and numeric

The integral panel does two separate things and reports them separately. It looks for an antiderivative in elementary form, and it computes the definite integral numerically. The second always works. The first works when the integrand is in the standard table, and says so plainly when it is not.

That distinction is deliberate and it is the opposite of what some tools do. A symbolic integrator that is right most of the time is worse than one that is right always and admits when it has nothing, because a student who could verify the answer would not have needed to ask. Where an antiderivative is found, it is checked by differentiating it back before it is shown; if the check fails, it is not shown at all.

The definite integral handles cases that catch simpler tools out. Improper integrals where the integrand is unbounded at an endpoint, such as the integral of 1/√x from 0 to 1, come out exactly right. So do integrals with infinite limits. Where the integrand has a pole inside the interval, you get a warning rather than a confident number that means nothing.

Underneath it all, the region is shaded on the graph — in one colour above the axis, another below — because a definite integral counts area below the axis as negative, and a student who has just got −2 for something that visibly encloses a region needs to see which part was subtracted.

Riemann sums

Left endpoints, right endpoints, midpoints, trapezoids and Simpson’s rule, with an adjustable number of subintervals and the rectangles drawn. This is the definition of the integral made visible: raise n and watch the approximation close on the exact value, and watch the midpoint rule beat both one-sided rules by an order of magnitude at the same n.

The sum is never used as the answer. It is shown next to the exact value so you can see the error, which is what a question about Riemann sums is normally asking about.

Curve analysis

Turning points, inflection points, and the intervals where the function is increasing, decreasing, concave up and concave down — each with the reasoning attached in the form a mark scheme wants. Not “minimum at (2, −1)” but “f′ = 0 and f″ = 2 > 0, so the curve is concave up here: a local minimum.”

Where the second derivative test is inconclusive — which happens whenever f″ is also zero at the critical point — the tool says so and falls back to the first derivative test rather than guessing. For f(x) = x³ at the origin it reports a stationary point that is neither a maximum nor a minimum, which is the truth; for f(x) = x⁴ it correctly finds a minimum despite f″(0) = 0.

A candidate inflection point is confirmed the same way. A zero of f″ is necessary but not sufficient: x⁴ has f″(0) = 0 and no inflection there, because the concavity does not actually change. Only points where the sign of f″ genuinely flips are reported.

Tangent and normal lines at any point come with the same panel, in point-slope form — which is the form the working usually needs — as well as slope-intercept.

Limits

One-sided and two-sided limits, with the table of values that defines them shown underneath. The table is not decoration: a limit is what the function does as you approach the point, and reading down a column of values at 0.1, 0.01 and 0.001 away is both the definition and what most homework questions on the topic are asking you to produce.

Where the limit exists but the function is undefined at the point, the tool says so — that is a removable discontinuity, a hole rather than a break. Where the one-sided limits differ, it says that too, and shows both.

How to type a function

The same parser as the graphing calculator, so anything that works there works here.

You want Type Notes
x squared x^2 x**2 and x² also work
3 times x 3x The multiplication sign is optional before a variable or bracket
Square root sqrt(x) Or √x, or x^(1/2)
Natural log ln(x) log(x) is base 10; log2(x) is base 2
e to the x e^x exp(x) is the same thing
Trig functions sin(x) Radians. asin, sec, sinh and the rest are all available
Constants pi, e Also π and tau. infinity works as an integration limit
Absolute value abs(x) Not differentiable at zero, and the tool will tell you so

One thing worth knowing: 2x^2 means 2·(x²), not (2x)², because powers bind tighter than multiplication everywhere in mathematics. If you meant the second, type (2x)^2.

The rules, and when each one applies

Every derivative this tool produces is one of these applied repeatedly. Knowing which is which is most of differential calculus.

Rule Shape Example
Power d/dx[xⁿ] = n·xⁿ⁻¹ d/dx[x⁵] = 5x⁴
Sum d/dx[f + g] = f′ + g′ d/dx[x² + x] = 2x + 1
Product d/dx[f·g] = f′g + fg′ d/dx[x·cos x] = cos x − x·sin x
Quotient d/dx[f/g] = (f′g − fg′)/g² d/dx[x/(x+1)] = 1/(x+1)²
Chain d/dx[f(g(x))] = f′(g(x))·g′(x) d/dx[sin 2x] = 2cos 2x
Exponential d/dx[eₕ] = eₕ·u′ d/dx[e³ₕ] = 3e³ₕ
Logarithm d/dx[ln u] = u′/u d/dx[ln(x²+1)] = 2x/(x²+1)
Logarithmic For fᵍ, where both base and exponent vary d/dx[xˣ] = xˣ(ln x + 1)

The one people reach for least often and need most is the chain rule. Almost every derivative that goes wrong in an exam goes wrong because an inner function was differentiated and then the factor was forgotten. If you find yourself getting cos(2x) where the answer is 2cos(2x), that is the one.

How the answers are simplified

A raw derivative is nearly unreadable. The product rule applied to x·cos(x) produces 1·cos(x) + x·(−sin(x)) before anything is tidied. Simplification collects like terms, cancels common factors, folds constants, and orders the result the way a textbook would — highest power first, coefficients in front.

One consequence is worth stating openly rather than leaving you to discover. Identical factors are cancelled between a numerator and a denominator, so x/x becomes 1. That is not an identity at x = 0, where the original expression is undefined and the simplified one is not. Every computer algebra system in common use does the same thing, for the good reason that refusing to cancel makes almost every quotient-rule answer unreadable — but it means a simplified result can have a slightly larger domain than what you typed.

In practice this matters when you are asked for the domain of a derivative as well as its formula. Work the domain out from the original function, not from the simplified answer.

On accuracy

The symbolic side is exact by construction: the rules are applied to a syntax tree, so there is no arithmetic to go wrong. What can go wrong is a mistyped rule, which is why every rule in the table is checked against numerical differentiation in the test suite — the symbolic derivative is compared with a central difference quotient at a spread of points, and a wrong rule fails immediately.

The numerical side takes more care. Definite integrals use double-exponential quadrature, which reaches machine precision on smooth integrands and, because it never evaluates the endpoints at all, handles improper integrals that make textbook methods return nothing. Where the integrand has a kink in the middle — |x| is the obvious case — it falls back to adaptive Simpson, which subdivides exactly where the error is.

Limits are estimated by evaluating closer and closer to the point, but not naively. Taking the value at the smallest step is wrong in a way that looks right: for (1 − cos x)/x², the numerator computes as exactly zero once x drops below about 10⁻⁸, so the sequence converges beautifully towards ½ and then collapses to 0. The estimate here stops at the step where the values stop settling and start being polluted by rounding, which is the last one you can trust.

Where a numerical answer cannot reach full precision, the tool says so and gives the uncertainty instead of printing digits it cannot stand behind.

Five mistakes this will catch

  1. A dropped chain-rule factor. Compare your answer with the one here at a single point. If yours is off by a constant multiple, this is why.
  2. The quotient rule the wrong way round. f′g − fg′ on top, not fg′ − f′g. The sign error is invisible in the algebra and obvious the moment you compare values.
  3. Calling a stationary point a minimum. Check whether f″ is actually non-zero there. If it is zero, the second derivative test says nothing at all.
  4. Ignoring the sign of an integral. If the shaded region includes area below the axis, the definite integral is not the total area. It is the signed area, and the panel shows you both colours for exactly this reason.
  5. Assuming a limit exists because the one-sided values look similar. Read down both columns of the table. Similar is not equal, and a jump of 10⁻⁵ is still a jump.

What it does not do

Worth knowing before you rely on it:

Questions

Why does my answer look different from the one shown here?

Usually because both are right and written differently. 2x/(2·sqrt(x²+1)) and x/sqrt(x²+1) are the same function. Substitute a couple of values into each — say x = 1 and x = 2 — and if the numbers agree everywhere you try, the expressions agree. If they disagree at even one value, one of them is wrong.

Why does the integral sometimes say it cannot find an antiderivative?

Because it genuinely has not, and saying so is more useful than guessing. It covers the standard table plus functions with a linear inner part. It does not attempt integration by parts on a product, partial fractions, or a substitution with a non-linear inner function. Some integrands, such as e^(-x²) and sin(x²), have no elementary antiderivative at all — no tool can produce one, because none exists. The definite value is still computed to full precision either way.

What does the tool mean when it says a result did not reach full precision?

Numerical integration of an awkward integrand has a limit. A function with an infinite tail that decays very slowly, or one that is unbounded at both endpoints, cannot be pinned down to fifteen digits in double-precision arithmetic. When that happens the tool tells you, and gives the uncertainty, rather than printing digits it cannot stand behind.

Is the second derivative test always conclusive?

No, and that is the point of how the classification is worded here. If f″ is zero at a critical point the test tells you nothing: x³, x⁴ and −x⁴ all have f′ = f″ = 0 at the origin, and they are a stationary inflection, a minimum and a maximum respectively. When f″ is zero, this tool falls back to checking the sign of f′ either side, which always decides.

Can I use this in an exam?

Almost certainly not. No web-based tool is permitted on the SAT, ACT or AP exams, and most university exams ban anything with internet access. Use it while you are learning the method and checking homework, and practise on whatever handheld calculator your exam allows.

Does it handle functions of variables other than x?

Yes. If your expression uses a single variable that is not x — t is common in physics problems — it differentiates with respect to that variable and labels the results accordingly. If more than one variable appears, it assumes x and treats the rest as constants, which is what a problem asking for dy/dx expects.

What about implicit differentiation and partial derivatives?

Neither yet. This tool differentiates an explicit function of one variable. Implicit differentiation of an equation such as x² + y² = 25, and partial derivatives of a function of several variables, are on the list.