Derivative Calculator

Type f(x). The derivative shows up as you type, steps included, every time.

Built by Bob Article by Lace QA by Ben Shipped September 15, 2026

How to use

1
Type your function in the f(x) box the way you'd write it on paper: 3x^2 + sin(x), x*e^x, (x^2+1)/(x-1). You can skip the multiplication sign (2x, xsin(x)) and the parentheses on simple arguments (sinx). The derivative updates a quarter second after you stop typing.
2
Pick an order. f′ is the first derivative, f″ the second, and so on up to f⁽⁵⁾. Higher orders repeat the whole process on the previous answer, and the steps show each intermediate derivative along the way.
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Optional: type a point in Evaluate at to get the slope there. A number, pi, e, or a constant expression like pi/4 or -2e all work. A decimal comma like 2,5 reads as 2.5, and 1,000 reads as one thousand.
4
Read the steps under How we got there. Each line names the rule it used and shows it with your terms. Copy derivative puts the plain-text answer on your clipboard, ready to paste into a graphing tool or your notes.

∫ Integral Calculator ➡️ Limit Calculator 📈 Graphing Calculator 📐 Slope Calculator

Frequently asked questions

How does the calculator find a derivative?

The same way you would on paper, just without the arithmetic slips. It reads your function into a tree, then walks it with the standard rules: the power rule for x^n, the product rule for f·g, the quotient rule for f/g, the chain rule whenever one function sits inside another, and the known derivatives of sin, cos, tan, ln, e^x and the rest. Each time a rule fires, it writes a step. Then it tidies the result: exact fractions instead of decimals, like terms combined, negative powers written as fractions. So x^-2 comes back as -2/x^3, not -2·x^(-2 - 1).

Why does the product rule step read f′g + fg′?

Because that's the rule. For y = f·g, the derivative is the derivative of the first factor times the second, plus the first times the derivative of the second. Take x·sin(2x). The first factor is x, whose derivative is 1, so the first half is sin(2x). The second half is x times the derivative of sin(2x), which needs the chain rule: cos(2x) times the derivative of 2x, or 2·cos(2x). Add them and you get sin(2x) + 2x·cos(2x). The steps show both halves with your own terms, so you can check which part you got wrong.

What does the value at a point tell me?

The slope of the curve at that x. For f(x) = x^2 the derivative is 2x, so at x = 3 the curve climbs 6 units for every unit you move right, and the tangent line there has slope 6. In physics, if f(t) is position, the first derivative at t is the velocity at that instant and the second derivative is the acceleration. In economics the derivative of a cost function is the marginal cost. Type the point in Evaluate at and the tool gives f′(a) to 6 significant figures.

Why does it say the derivative is undefined at my point?

There are two honest reasons. First, if f itself isn't defined there, neither is its slope. ln(x) at x = -1 is the classic trap: the formula 1/x happily returns -1, but ln(-1) doesn't exist for real numbers, so there's no curve at that point to have a slope. We check f first. Second, the curve can have a corner or a vertical tangent. abs(x) at 0 has slope -1 on the left and 1 on the right, so no single slope works. x^(1/3) at 0 goes straight up. In both cases the derivative doesn't exist, and we say so instead of printing a number.

What does "Read from the slope on both sides" mean?

Sometimes the formula breaks at a point where the derivative still exists. abs(x)^3 is smooth at 0, but its derivative formula has abs(x) in a denominator, so plugging in 0 gives 0/0. When that happens, and f is defined at the point, we measure the slope directly: the rise over the run from the left and from the right, with smaller and smaller steps. If both sides settle on the same number, that's the derivative, and the note tells you it came from the numbers rather than the formula. If they disagree, the derivative doesn't exist there.

What are second and higher derivatives for?

The second derivative tells you how the slope itself is changing. Where f″ is positive the curve bends upward like a cup, where it's negative it bends down, and where it changes sign you have an inflection point. That's the second derivative test for telling a minimum from a maximum. In motion problems, the second derivative of position is acceleration and the third is jerk. The fourth and fifth show up in Taylor series, where the nth derivative at a point sets the nth coefficient.

Why doesn't my answer look exactly like the one in my textbook?

Derivatives have many equal forms, and textbooks differ. We keep products of sums the way the rule produced them instead of multiplying them out, because that's closer to the working and easier to check. So the derivative of (x^2+1)/(x-1) reads (2x(x - 1) - (x^2 + 1))/(x - 1)^2, straight from the quotient rule. Your book might expand the top to x^2 - 2x - 1. Both are right. To confirm two forms match, evaluate both at a couple of points; if they agree at several x values, they are almost certainly the same function.

How do I type functions?

Use + - * / ^ and parentheses, with x as the only variable. Constants: pi and e. Functions: sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin works too), sqrt, exp, ln for the natural log, log for base 10, and abs. Powers can be negative or fractional: x^-2, x^(1/3). Implicit multiplication works, so 2x, 3sin(x) and x(x+1) all read the way you'd expect. Letters other than x, like a or k, aren't supported yet.

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What is a derivative?

A derivative tells you how fast something changes. Give it a function, and it hands back a new function whose value at any x is the slope of the original curve at that point. Type x^2 into the Derivative Calculator and you get 2*x: at x = 3 the curve is climbing 6 units for every unit you move right.

Newton and Leibniz worked out the idea separately in the late 1600s, and they argued about who got there first for the rest of their lives. We still use both of their notations. Newton's dot and Lagrange's prime (f′) mean the same thing as Leibniz's dy/dx.

Derivatives show up far outside the calculus classroom. Velocity is the derivative of position. Acceleration is the derivative of velocity. Marginal cost is the derivative of a cost curve. Any time someone asks "how quickly is this changing right now?", a derivative is the answer.

A lot of online math solvers give you the answer for free and charge for the how. The steps sit behind a Pro badge, a sign-in wall, or a daily ration that runs out halfway through the problem set. This derivative calculator shows the answer and every step, every time, and then lets you leave. The math runs in your browser, so nothing you type goes anywhere.

How to use the derivative calculator

Type your function in the f(x) box the way you'd write it on paper: 3x^2 + sin(x), x*e^x, (x^2+1)/(x-1). You can skip the multiplication sign (2x, xsin(x)). The only variable is x.
Watch the answer appear. There's no Calculate button. The result updates a quarter second after you stop typing.
Pick an Order if you need more than the first derivative. f′ is the first, f″ the second, and so on up to f⁽⁵⁾.
Optional: type a point in Evaluate at to get the slope there. A number, pi, e, or a constant expression like pi/4 all work. A decimal comma like 2,5 reads as 2.5.
Read the steps under How we got there. Each line names the rule and shows it with your own terms.
Hit Copy derivative to grab the plain-text answer for your notes or a graphing tool.

The page opens with 3x^2 + sin(x) already filled in, so you see 6*x + cos(x) and three steps before you touch anything. That's the whole flow: open, type, read, leave.

The rules behind every derivative

You don't need a limit every time you differentiate. A short list of rules covers almost every function you'll meet in a first calculus course, and the calculator uses exactly these:

Power rule: (xⁿ)′ = n·xⁿ⁻¹
Product rule: (f·g)′ = f′g + fg′
Quotient rule: (f/g)′ = (f′g − fg′) / g²
Chain rule: f(g(x))′ = f′(g(x)) · g′(x)

Add the known derivatives of sin, cos, tan, eˣ, ln and friends, and you can take apart almost anything. The trick is knowing which rule to reach for first. That's also where most mistakes happen, which is why the steps matter more than the final line.

Worked example: x·sin(2x) at x = 1

Type x*sin(2x) and set Evaluate at to 1. The calculator returns sin(2*x) + 2*x*cos(2*x) and two steps:

Product rule: (x*sin(2*x))' = (x)'*sin(2*x) + x*(sin(2*x))'. The derivative of x is 1, so the first half is just sin(2*x).
Chain rule: (sin(2*x))' = cos(2*x)*(2*x)' = 2*cos(2*x). The outer function is sin, the inner is 2x, and the inner derivative 2 comes out front.

Put the halves together and you get sin(2*x) + 2*x*cos(2*x). At x = 1 that's sin(2) + 2·cos(2), or about 0.9093 − 0.8323. The value box shows 0.0770038. The curve is almost flat there, rising just under 0.08 units per unit of x.

If your homework answer is sin(2x) + x·cos(2x), the chain rule step shows exactly where the missing 2 went.

Common derivatives, checked against the calculator

Every row below was run through the calculator. The middle column is the exact string Copy derivative puts on your clipboard, and the last column is the rule name you'll see in the steps.

You type	Derivative	Rule in the steps
`x^5`	`5*x^4`	Power rule
`x^-2`	`-2/x^3`	Power rule
`x^(1/3)`	`1/(3*x^(2/3))`	Power rule
`sqrt(x)`	`1/(2*sqrt(x))`	Root rule
`e^x`	`e^x`	Exponential rule
`2^x`	`2^x*ln(2)`	Exponential rule
`ln(x)`	`1/x`	Log rule
`log(x)`	`1/(x*ln(10))`	Log rule
`cos(x)`	`-sin(x)`	Trig rule
`tan(x)`	`sec(x)^2`	Trig rule
`asin(x)`	`1/sqrt(1 - x^2)`	Inverse trig rule
`atan(x)`	`1/(1 + x^2)`	Inverse trig rule
`sin(x^2)`	`2xcos(x^2)`	Chain rule
`(3x+1)^4`	`12(3x + 1)^3`	Chain rule
`x^2*e^x`	`2xe^x + x^2*e^x`	Product rule
`x^x`	`x^x*(ln(x) + 1)`	Logarithmic differentiation

Two rows trip people up. log(x) means base 10 here, so its derivative carries a ln(10) in the denominator. Use ln(x) if you mean the natural log. And x^x fits neither the power rule nor the exponential rule, because x sits in both the base and the exponent. The calculator switches to logarithmic differentiation and says so.

Notice what the answers don't look like. You won't see -2*x^(-2 - 1) or x/x left lying around. Exponents fold to exact fractions, negative powers print as quotients, and like terms combine. Type sin(x)^2 + cos(x)^2 and the answer is a clean 0, the way it should be.

How to find a maximum with the second derivative

The first derivative tells you where a curve is flat. The second derivative tells you whether that flat spot is a peak or a valley. Here's the full test on f(x) = x³ − 6x² + 9x.

Type x^3 - 6x^2 + 9x with order f′. You get 3*x^2 - 12*x + 9, which factors to 3(x − 1)(x − 3). Evaluate at 1 and at 3: both give 0. Those are the two flat spots.

Now switch the order to f″. The answer is 6*x - 12, and the steps show f′ as an intermediate line before differentiating again. Evaluate at 1 and you get -6. Negative means the curve bends downward, so x = 1 is a local maximum, at the point (1, 4). Evaluate at 3 and you get 6. Positive means it bends upward, so x = 3 is a local minimum, at (3, 0).

Keep going to f‴ and you get 6. At f⁽⁴⁾ the answer is 0. A cubic runs out of derivatives after three, and the calculator shows it happening.

Tip: for motion problems, type position with x standing in for time. -4.9x^2 + 20x is a ball thrown up at 20 m/s. Its first derivative at 2 is 0.4, so the ball is still rising, barely. The second derivative is a constant -49/5, which is −9.8 m/s², gravity.

Edge cases and limitations

Some points have no derivative, and a good calculator should say so instead of printing a number.

The function isn't defined at your point. ln(x) at -1 is the classic trap. The formula 1/x would happily return −1, but ln(−1) doesn't exist, so there's no curve to have a slope. The calculator checks f first and reports that f′ is undefined there.
Corners and cusps. abs(x) at 0 has slope −1 on the left and 1 on the right, so no single slope works. x^(1/3) at 0 goes straight up. Both come back undefined.
The formula breaks but the slope exists. abs(x)^3 at 0 gives 0/0 in its formula, yet the curve is smooth there. The calculator measures the slope from both sides, gets 0, and adds a note saying the value came from the numbers rather than the formula.
Huge values. e^x at 1000 is too big for any computer number. You get a message asking for a smaller x.

The scope is deliberate: one variable, x, up to the 5th derivative. Letters like a or k aren't supported yet, and neither are sinh, cosh or tanh. Partial derivatives and implicit differentiation are different jobs and aren't part of this tool.

Related calculations

Integration runs the derivative backwards. The Integral Calculator reads the same function syntax, so you can check an antiderivative by differentiating it here.

Derivatives are defined as limits. The Limit Calculator shows what happens as x approaches a value, which is handy when a formula hits 0/0.

Seeing the curve helps. Paste your function into the Graphing Calculator and check that the slope you computed matches what the picture shows. For the slope of a straight line between two points, the Slope Calculator is the simpler tool. And when f′ turns out to be a quadratic, like 3*x^2 - 12*x + 9 above, the Quadratic Formula Calculator finds where it hits zero.

Frequently asked questions

Why is the derivative of e^x just e^x?

Because e is the one base where the growth rate equals the current value. Any exponential aˣ has derivative aˣ·ln(a), and ln(e) = 1, so the extra factor disappears. Try 2^x and you get 2^x*ln(2). Set e^x to order 5 and the answer is still e^x.

What's the difference between f′(x), dy/dx and y′?

Nothing, as far as the answer goes. They're three notations for the same derivative. f′(x) comes from Lagrange, dy/dx from Leibniz, and y′ is shorthand when y = f(x). The calculator uses primes: f′, f″, f‴, then f⁽⁴⁾ and f⁽⁵⁾.

Can it do partial derivatives or implicit differentiation?

No. This tool differentiates functions of a single variable, x. Partial derivatives need several variables, and implicit differentiation needs an equation in x and y. Both are separate jobs, and a derivative calculator that tries to do everything tends to make each part harder to use.

Why did my decimal turn into a fraction?

The calculator does exact arithmetic, so 4.9 becomes 49/10. The derivative of -4.9x^2 + 20x comes back as -(49/5)*x + 20, which is −9.8x + 20. It's the same number, and it never drifts into rounding noise like 9.800000000000001. Values at a point still print as decimals.

Is it cheating to use a derivative calculator for homework?

It depends on how you use it. Copying the last line teaches you nothing. Doing the problem first, then comparing your working against the steps, is how you find the exact line where you dropped a sign or forgot the chain rule. Paywalled steps make that second habit expensive. Here they're just there.

Does it work on my phone?

Yes. It's one card with three inputs, so it fits a phone screen, and the math runs locally in the browser. Nothing is sent to a server, so a weak signal doesn't slow it down once the page has loaded.

Derivative Calculator

How we got there

How to use

Related tools

Frequently asked questions

Ratings & Reviews

Rate this tool

What is a derivative?

How to use the derivative calculator

The rules behind every derivative

Worked example: x·sin(2x) at x = 1

Common derivatives, checked against the calculator

How to find a maximum with the second derivative

Edge cases and limitations

Related calculations

Frequently asked questions

Why is the derivative of e^x just e^x?

What's the difference between f′(x), dy/dx and y′?

Can it do partial derivatives or implicit differentiation?

Why did my decimal turn into a fraction?

Is it cheating to use a derivative calculator for homework?

Does it work on my phone?