Derivative Calculator

How to Use the Derivative Calculator

Type your function

Enter f(x) using ^ for powers and function names like sin, cos, ln, exp, and sqrt — for example x^2*sin(x). You can write 2x for 2 times x.

Set the variable

The variable defaults to x, but you can differentiate with respect to any letter — t, θ written as a letter, and so on.

Read the derivative

The calculator returns f′(x) in simplified form and names which rules it used — power, product, quotient, or chain — so you can check your own working.

Evaluate at a point

Type a value into "At" to get the slope of the tangent there — the instantaneous rate of change at that exact point.

Differentiation, Done For You

The Rate of Change

The derivative of a function measures how fast it is changing — the slope of its graph at every point. If a function gives your position over time, its derivative gives your speed; if it gives the cost of producing goods, the derivative gives the marginal cost of one more. This idea, the instantaneous rate of change, is the foundation of calculus and one of the most consequential concepts in all of mathematics and science. Computing a derivative by hand means applying a small set of rules: the power rule (the derivative of xⁿ is n·xⁿ⁻¹), the product rule, the quotient rule, and the chain rule for functions nested inside functions, plus the known derivatives of the standard functions like sine, cosine, exponential, and logarithm. This calculator applies all of those automatically and gives you the simplified result, then tells you which rules were involved so you can follow the logic rather than just copy an answer.

What is worth knowing is how this calculator works under the hood, because it is unusual. Many online derivative tools either send your function to a remote maths engine or rely on a large third-party library. This one does the differentiation itself, with a small, purpose-built symbolic engine that runs entirely in your browser: it parses your function into a tree of operations, applies the differentiation rules to that tree exactly the way the rules are stated in a textbook, and then simplifies the result. Because it is deterministic and rule-based — not a numerical approximation — the answer is the exact symbolic derivative, the same expression you would get by hand, just without the slips. And because nothing leaves your device, you can differentiate freely and privately, even offline.

"A derivative is the slope of a curve at a single point — speed from position, marginal cost from cost. It is the question 'how fast is this changing right now?' made precise."

From Slopes to the Real World

Derivatives are not an academic curiosity; they are how change is modelled everywhere. Physics is written in derivatives — velocity is the derivative of position, acceleration the derivative of velocity, and the whole of mechanics follows. Economics uses them for marginal cost and revenue; biology for growth rates; engineering for optimisation, since the highest and lowest points of a curve occur exactly where its derivative is zero. That last use is why "set the derivative to zero" is the universal recipe for finding a maximum or minimum, from the cheapest design to the best dose. Evaluating the derivative at a specific point — which this tool also does — gives the slope of the tangent line there, the best straight-line approximation to the curve at that instant, which underlies everything from Newton's method to the way machine-learning models learn by gradient descent. Whether you are checking calculus homework, exploring a function's behaviour, or just curious how something changes, this calculator gives the exact derivative and its value at any point, instantly and privately.

10 Facts About Derivatives

01

A derivative is the slope of a function at a point — its rate of change.

02

The power rule: the derivative of xⁿ is n·xⁿ⁻¹.

03

The chain rule differentiates a function nested inside another.

04

Velocity is the derivative of position; acceleration of velocity.

05

A function's maximum or minimum occurs where its derivative is zero.

06

The derivative of eˣ is eˣ — it is its own derivative.

07

Newton and Leibniz developed calculus independently in the 1600s.

08

This tool computes the exact symbolic derivative, not an approximation.

09

Gradient descent — how AI models learn — is built on derivatives.

10

Everything runs in your browser — no server, no library upload.

Frequently Asked Questions

Use ^ for powers, * for multiplication (or just write 2x), and function names like sin, cos, tan, ln, exp, and sqrt with brackets — for example x^2*sin(x) or sqrt(x^2+1). The calculator differentiates whatever you type and simplifies the result.
All the standard ones: the power rule, product rule, quotient rule, and chain rule, plus the known derivatives of sine, cosine, tangent and their inverses, the exponential and logarithm, and square root. After computing the derivative it tells you which of these rules were applied, so you can compare with your own working.
Exact. The calculator computes the symbolic derivative — the actual formula — by applying the rules to the structure of your function, not by numerical approximation. It then simplifies it. The result is the same expression you would get differentiating by hand, just without arithmetic slips.
It gives the value of the derivative at that input — which is the slope of the tangent line to the curve there, the instantaneous rate of change at that exact point. For instance, evaluating the derivative of position at a time gives the speed at that moment.
Yes — change the "Variable" box to any letter, such as t. Every other letter in the expression is then treated as a constant, which is exactly how partial-style single-variable differentiation works.
Because a smooth function reaches its maximum or minimum exactly where its slope is zero — the tangent is flat there. So setting the derivative to zero and solving is the standard method for optimisation: finding the cheapest, fastest, or best value. Differentiate here, then solve the result with the Factoring or Quadratic tools.
Yes — that is one of its core abilities. For something like sin(x²) it differentiates the outer function and multiplies by the derivative of the inside, giving 2x·cos(x²). It nests the chain rule to any depth.
No. The differentiation happens entirely in your browser using a small built-in symbolic engine — nothing is sent to a server or a third-party library. It works offline once the page has loaded.
It shows a clear message asking you to check the syntax — usually a missing bracket or an unsupported name. Stick to ^ for powers, explicit brackets after function names, and the standard functions, and it will parse correctly.
Completely free, with no account, sign-up, or limit. It runs entirely in your browser and collects no data. Differentiate as many functions as you like.

Related News

You may be interested in these recent stories from our newsroom.

No related news yet for this tool. Our editorial team publishes new pieces every week.

Browse all news →