Derivative calculator that shows the rules behind each answer
Type the function in the top row, pick the variable and the order, and the row below fills in with the derivative, the rules that were used, the tangent slope at the point you enter, and a graph of the function beside its derivative. There is no calculate button; editing the function recalculates everything.
Notation · ln(x) natural log · log(x) base-10 log · 2x times sign optional · x^2 power · 0.5 decimal point in the formula · sqrt(x) square root
- f(a) value
- — The original function at the point you entered
- f'(a) tangent slope
- — Slope of the tangent line through that point
- f''(a) concavity
- — Positive is concave up, negative is concave down
- Tangent line
- — The line that touches the curve at that point
| Function | Expression | Value |
|---|
This calculator parses and differentiates your expression in the browser and never downloads an external computation library. While tidying the answer it cancels factors shared by the numerator and denominator, and that cancellation assumes the shared factor is not zero. Functions that fail to be differentiable at a point, such as the absolute value at zero, get a derivative valid on the rest of the domain. Trigonometric identities are not applied, so an answer can look different from your textbook while still being equal.
ready to use.
- Visible firstKeep the input and result positions clear.
- Results firstPut the main number up front and keep the process secondary.
- Less to askNo sign-up or extra information before using the tool.
What a derivative actually tells you, and how this calculator gets there
A derivative reports slope, not height. How high the curve sits at a point is what the function value answers; how steeply it climbs there is what the derivative answers. On an AP or calculus exam, "find the equation of the tangent line" and "find where the function is increasing" are the same computation wearing different clothes, and both start with one derivative. The part students actually get stuck on is deciding which rule applies where.
So this calculator does not just hand back an answer. It lists the rules it walked through to build it. Enter x^2*sin(x) and you see the product rule, the power rule and trigonometric differentiation appear as tags; enter sqrt(3x^2+1) and you see the radical rule and the chain rule instead. The sections below explain how the parser reads your typing, why ln and log are kept apart, and why second derivatives of quotients grow so long, using the exact values the screen produces.
Start with one polynomial and follow it all the way down
Enter f(x) = x^3 + 2x^2 - 5x + 3 and set the evaluation point to 2. The screen fills in the values in the table below. The first derivative comes from applying the power rule term by term, and the tangent slope is that derivative evaluated at 2. The table uses the same rounding the tool displays.
- Power rule: differentiating x^n gives n·x^(n-1), so x^3 becomes 3x^2 and 2x^2 becomes 4x.
- The constant term 3 differentiates to 0 and disappears from the answer.
- The term -5x keeps only its coefficient and becomes -5. That is the constant multiple rule.
- The tangent slope is f′(2) = 3·4 + 4·2 - 5 = 15.
- The tangent line takes that slope and shifts to pass through the point, giving y = 15x - 21.
| Item | Value | Where it comes from |
|---|---|---|
| First derivative f′(x) | 3x^2 + 4x - 5 | Power rule and constant multiple rule, term by term |
| Second derivative f″(x) | 6x + 4 | The first derivative differentiated again |
| Third derivative f‴(x) | 6 | The second derivative differentiated again |
| Function value f(2) | 9 | 8 + 8 - 10 + 3 |
| Tangent slope f′(2) | 15 | 3×4 + 4×2 - 5 |
| Concavity f″(2) | 16 | 6×2 + 4, positive so concave up |
| Tangent line | y = 15x - 21 | 9 - 15×2 = -21 |
How to type an expression so the parser reads what you mean
Two things trip people up: the multiplication sign and the logarithm base. This calculator lets you drop the multiplication sign, and it reads a bare log as base 10, which matches how American precalculus and calculus texts print it. Natural log is ln. The notation row under the input keeps that agreement on screen.
- 2x, 3sin(x), x(x+1) and (x+1)(x-1) are all read as products.
- Exponents use the caret: x^2, x^(1/2) and 2^x all work.
- ln(x) is the natural log, log(x) is base 10, log2(x) is base 2.
- Roots are sqrt(x) and cbrt(x); x^0.5 means the same as sqrt(x).
- Inside the formula the decimal separator is a period, because a comma collides with the argument separator used in international formula notation.
- Write pi or π for π, and e for Euler’s number.
Why the rules are shown next to the answer
Half the people who look up a derivative calculator already have an answer written down. What they lack is confidence that they picked the right rule. This tool collects the rules it used while walking the expression and lists them under the result. If your answer matches but the tags do not match your work, there was more than one road to the same place.
- The constant rule and the identity rule are too basic to tag, so they only show up when the whole function is constant.
- The sum and difference rule appears whenever there is more than one term.
- The constant multiple rule replaces the product rule when one factor does not involve the variable, so the 3 in 3sin(x) simply rides along.
- The chain rule only appears when the inside of a function is something other than the bare variable: sin(x) has none, sin(3x+1) does.
- Logarithmic differentiation shows up when the variable appears in both the base and the exponent, as in x^x.
Seeing exactly where the chain rule enters
The chain rule says to differentiate the outside and multiply by the derivative of the inside. That sentence is easy; spotting where the inside stops is not. Press the chain rule example, sqrt(3x^2+1), and the answer is 3x / sqrt(3x^2+1). The 3x in the numerator is what is left after multiplying the inside derivative 6x by the outside derivative 1/(2√u).
- sin(3x+1) differentiates to 3cos(3x+1); the leading 3 is the inside derivative.
- exp(2x) differentiates to 2exp(2x) for the same reason.
- ln(x^2+1) differentiates to 2x / (x^2+1), where the numerator is the inside derivative.
- (3x+1)^4 differentiates to 12(3x+1)^3 because 4 and 3 multiply.
- When the inside is just x, the chain rule tag disappears, since the factor it contributes is 1.
Choosing between the product rule and the quotient rule
A fraction bar makes people reach for the quotient rule reflexively, but if the denominator holds no variable the quotient rule is wasted effort. The calculator makes the same judgement: a constant denominator is handled as a constant multiple, which keeps the answer short, and the quotient rule tag only appears when the denominator actually varies.
- x^2/5 has a constant denominator, so it finishes as (1/5)·2x = 0.4x.
- (x^2+1)/(x-1) needs the quotient rule and returns (x^2 - 2x - 1)/(x-1)^2.
- 4/x^2 is faster as 4x^(-2): one power rule gives -8/x^3.
- Rewriting a single-term denominator with a negative exponent saves real time on a timed exam.
- The product rule is only necessary when both factors contain the variable, as in x^2·sin(x).
What the evaluation point adds
A symbolic derivative does not answer "how steep is it right here". Enter a point and four result cards report the state of the curve there: the function value, the tangent slope, the second derivative, and the tangent line itself. Leave the point blank and the faint example value is used instead, with a note above the cards saying so.
- f(a) is the height of the curve, which is the y-coordinate the tangent line has to pass through.
- f′(a) is the tangent slope. Positive means increasing at that point, negative means decreasing.
- f′′(a) is concavity: positive is concave up, negative is concave down.
- The tangent line is built as y = f′(a)·x + (f(a) - f′(a)·a).
- Feed it a point where the denominator vanishes and the card reads undefined instead of a number.
Reading f and its derivative on the same axes
Overlaying the two curves makes the relationship visible. The dashed curve is the original function and the solid curve is the derivative of the order you selected. Wherever the solid curve crosses the horizontal axis, the dashed curve has a horizontal tangent, which is a candidate for a maximum or a minimum. The axes, grid and legend are drawn before you type anything, so you can see the window in advance.
- The horizontal window spans 5 units either side of the evaluation point.
- The vertical window is chosen from the spread of the sampled values, and extreme values near an asymptote are clipped so the rest of the plot stays readable.
- The vertical dotted line marks the evaluation point; the circle on it is f(a) and the square is the derivative value.
- Where the solid curve sits above the axis the dashed curve rises, and where it sits below the dashed curve falls.
- Switch the order to 2 and the solid curve becomes the second derivative, which shows which way the curve bends.
When the second and third derivatives earn their keep
The first derivative decides increasing or decreasing. The second decides whether that change is speeding up or easing off, which is what concavity and inflection points are about. In physics, differentiating position with respect to time gives velocity and differentiating again gives acceleration; the third derivative, jerk, turns up in ride-comfort engineering.
- A point where f′ = 0 and f′′ > 0 is a local minimum.
- A point where f′ = 0 and f′′ < 0 is a local maximum.
- A point where f′′ changes sign is an inflection point.
- s(t) = t^3 - 6t^2 + 9t gives velocity 3t^2 - 12t + 9 and acceleration 6t - 12.
- Setting the order to 3 puts f through f‴ in the table together.
Why a second derivative suddenly looks enormous
Differentiating a quotient or a radical twice makes the expression visibly longer. Each pass through the quotient rule squares the denominator, and the product rule piles new terms into the numerator. This calculator cancels the factors that survive on both sides and folds constants, but it does not attempt trigonometric identities or factoring.
- sqrt(3x^2+1) has the short first derivative 3x / sqrt(3x^2+1), yet its second derivative already carries two numerator terms.
- ln(sin(x)) returns (-sin(x)^2 - cos(x)^2) / sin(x)^2, which the Pythagorean identity would write as -1/sin(x)^2. The values agree.
- When your form differs from the textbook, substitute two or three arbitrary values into both and check they agree.
- The longer the expression, the easier it is to drop a sign while copying it by hand, which is what the copy button is for.
Input mistakes that come up again and again
When an expression cannot be read, the results clear and the status row says which character position stopped the parser. Almost every case is an unclosed bracket or a function name without brackets. These are the forms that show up most often.
- sin x — a function name needs brackets. Write sin(x).
- sin(x — the closing bracket is missing, and the status row points near the offending character.
- x^2^3 — stack exponents inside brackets so the meaning is unambiguous: x^(2^3).
- 2**x — a double asterisk is not an exponent operator here. Write 2^x.
- sin^2(x) — an exponent on the function name is not read. Write sin(x)^2.
- 1,5 — a comma is not a decimal point inside a formula. Write 1.5.
What this calculator does not do
Its job ends at the derivative of one variable. The tasks below need a different method, and treating a result here as the answer to one of them will mislead you.
- Integration: the reverse direction is not computed.
- Implicit differentiation: a relation such as x^2 + y^2 = 1 is not handled. Solve for y first.
- Multivariable notation: with more than one letter present, everything except the chosen variable is treated as a constant. The result matches a partial derivative, but partial notation is not supported.
- Piecewise functions: enter each branch separately.
- Detecting non-differentiable points: abs(x) has no derivative at 0, but the printed derivative simply excludes that point.
- Numerical approximation: functions that cannot be written with elementary functions are out of scope.
Check these before you copy the answer onto your paper
Running through this short list catches most of the marks people lose on derivative questions.
- Is the differentiation variable the one the question asked for? The menu is filled from the letters in your expression.
- Is the order the one the question wants? It starts at 1.
- If you meant natural log, did you type ln rather than log?
- Does a cancelled factor vanish anywhere inside the domain the question specifies?
- If the question asks for a tangent line, did you actually enter the point rather than leaving the example value?
- For a long expression, did you use the copy button instead of transcribing it?
Sources and review date
Content review date for this page: 2026-09-05. Differentiation rules are mathematical definitions rather than policy that changes each year, so nothing here goes stale. The sources below are the standard references this calculator was checked against.
Related tools
Derivative calculator questions
There is no calculate button. When does it run?
It runs as you type. Every keystroke in the function box refreshes the derivative, the rule tags, the point cards, the table and the graph together, and the same happens when you change the variable or the order. A calculate button would add a click without changing any answer, so it was removed. While the expression is still incomplete, for example sin(x with no closing bracket, the status row tells you where the parser stopped instead of showing a result.
I entered log(x) and the derivative is not 1/x
In this calculator log means the base-10 logarithm, so log(x) differentiates to 1 / (x·ln(10)). For the natural logarithm type ln(x), which differentiates to 1/x. US precalculus and calculus texts print log for base 10 and ln for the natural log, and the same convention holds in Korean, Spanish and Indonesian textbooks, so that is the convention the tool uses in every language.
Do I have to type the multiplication sign?
No. 2x, 3sin(x), x(x+1) and (x+1)(x-1) are all read as products. Variables are single letters, so xy means x times y and ab means a times b; multi-letter variable names are not available. Function names are the exception: only the registered names such as sin, cos, exp and ln are read as multi-letter tokens, and each of them must be followed by brackets.
Which functions are supported?
Polynomials and rational expressions, square and cube roots, the six trigonometric functions (sin, cos, tan, sec, csc, cot), three inverse trigonometric functions (asin, acos, atan), three hyperbolic functions (sinh, cosh, tanh), exponentials (exp and a^x), logarithms (ln, log, log2) and the absolute value. Spanish spellings such as sen, arcsen, tg and ctg are accepted, as are shorthand forms like log10 and lg.
Can it integrate as well?
No, it only goes in the differentiation direction. Finding a derivative and recovering an antiderivative are very different problems: differentiation always terminates by following rules, while many perfectly ordinary functions have no antiderivative expressible with elementary functions. You can, however, differentiate repeatedly, which is what the order control does up to the third derivative.
Can I differentiate with respect to t instead of x?
Yes. The variable menu is filled from the single letters that appear in your expression. Enter s(t) = t^3 - 6t^2 + 9t, pick t, and you get the velocity 3t^2 - 12t + 9. When several letters appear, only the selected one is treated as a variable and the rest are constants, so differentiating a·x^2 with respect to x returns 2ax.
The second derivative is far longer than my textbook answer
Differentiating a quotient or a radical twice squares the denominator and stacks terms in the numerator. The calculator cancels shared factors and folds constants, but it does not apply trigonometric identities or factor the result. For instance ln(sin(x)) returns (-sin(x)^2 - cos(x)^2) / sin(x)^2 where the identity form is -1/sin(x)^2. The two agree everywhere they are defined; substitute a couple of values into both if you want to confirm it.
What happens if I leave the evaluation point empty?
The faint example number is used for the calculation, and a line above the result cards says the value is an example rather than something you typed. As soon as you enter your own number that note changes, and the tangent line and the vertical marker on the graph move with it. Decimals may be typed with either a period or a comma in that box, so 1.5 and 1,5 are read the same way.
How do I enter sin squared x?
Write sin(x)^2. The handwritten form sin^2(x) is not read, because the exponent has to be attached to something the parser can see. For the same reason sin(x^2) and sin(x)^2 are different functions, so check the bracket position: the first one picks up the chain rule and the second picks up the power rule.
What does the copy button put on the clipboard?
It copies the derivatives up to the order you selected together with the function value, the tangent slope and the tangent line at your point, one item per line. The expressions are written with an asterisk for multiplication rather than the middle dot shown on screen, so the text can be pasted straight back into this calculator or into other mathematical software and still parse. It exists mainly to stop sign errors during transcription.