Derivative Calculator

This derivative calculator finds the derivative of any function you type in, and instead of just handing you a final answer, it walks through the working so you can see how that answer was reached. It covers the range of differentiation cases you would run into in a typical calculus course: applying the correct rule, taking a second or third order derivative, evaluating the result at a specific point, and plotting the function alongside its derivative. Enter a function below and select Solve to get started.

What Does This Derivative Calculator Do?

At its core, the calculator takes a function of x, or whichever variable you are working with, and returns f'(x), showing which rule was applied to reach it. It is built to handle polynomial, trigonometric, exponential, logarithmic, and rational functions without you needing to specify which type you are dealing with, since it works that out from the input itself. Alongside the algebraic result, the same page plots the original function so you can check the derivative visually, comparing where the curve rises, falls, or flattens against what the slope function shows.

How Do You Use the Derivative Calculator?

1. Type your function into the input box, using x as the default variable unless your problem calls for another one.

2. Select the variable you are differentiating with respect to, if your function involves more than one.

3. Choose the order of the derivative you need: first, second, third, and so on.

4. Select Solve, and the calculator returns the derivative along with the steps it took to reach it.

Most of the confusion at this stage has nothing to do with calculus itself, it comes from how the input is typed. Implicit multiplication, such as 5x, needs to be written in a way the parser recognizes, trig and log functions need their parentheses (sin(x), not sinx), and compound expressions need to be grouped correctly or the calculator will read the function differently than you intended.

What Input Format Does the Calculator Accept?

A few small formatting habits account for most of the errors people run into on the first try. Powers use the caret symbol, multiplication is either written out or left implicit depending on the term, and trig and log functions always need their argument wrapped in parentheses. A couple of quick examples make this clearer than a list of rules would:

Which Differentiation Rules Does the Calculator Apply?

Behind every result, the calculator is choosing from a fixed set of rules, the same ones you would reach for if solving by hand: the power rule, the sum and difference rule, the product rule, the quotient rule, and the chain rule. What the tool actually saves you from is the identification step, working out which rule, or combination of rules, a given expression calls for, which is usually where mistakes creep in when solving manually. The sections below walk through each rule on its own.

What Is the Power Rule in Derivative Calculation?

The power rule states that the derivative of x^n is n times x^(n-1). It is the rule used most often, since it applies to every polynomial term individually. Take 4x^5 - 3x^2 + 7x - 10: differentiating term by term gives 20x^4 - 6x + 7.

What Is the Product Rule in Derivative Calculation?

The product rule applies whenever two functions of x are multiplied together rather than added or subtracted. It states that the derivative of u times v equals u' times v, plus u times v'. For (x^2 - 5)(x^3 - 2x + 3), the calculator applies this rule directly rather than expanding the expression first.

What Is the Quotient Rule in Derivative Calculation?

When one function is divided by another, the quotient rule takes over: the derivative of u/v equals (u'v - uv') divided by v squared. For a function such as (x^2 + 1)/(5x - 3), it is easy to reverse the order in the numerator by mistake, subtracting u'v instead of uv', and that mix-up is the single most common error students run into with this rule.

What Is the Chain Rule in Derivative Calculation?

The chain rule handles composite functions, where one function sits inside another. For f(g(x)), the derivative is f'(g(x)) times g'(x). Take (4x^3 - 2x)^6: the outer function is something raised to the sixth power, and the inner function is 4x^3 - 2x. Spotting that split correctly is where most manual attempts go wrong, and it is the exact step the calculator flags automatically.

What Are the Common Derivatives for Trig, Log, and Exponential Functions?

A handful of standard derivatives come up often enough that it helps to have them in one place, as a quick lookup rather than something to re-derive each time:

Can the Calculator Find Second, Third, or Higher-Order Derivatives?

Yes. The tool is not limited to a single pass, you can request the second derivative, the third, or go further depending on what the problem needs. A simple way to picture what a higher-order derivative actually represents is the position-velocity-acceleration chain from physics: the derivative of position gives velocity, and the derivative of velocity gives acceleration. To choose the order, use the setting in the calculator before selecting Solve.

Can the Calculator Evaluate a Derivative at a Specific Point?

Finding f'(x) gives you the general formula, but sometimes what you actually need is a single number, the slope at one particular point. The calculator handles that too: enter a value for x alongside your function, and it plugs that value into the derivative to return the slope of the tangent line at that exact spot. This is the step most people need for tangent line problems, or for optimization problems where the goal is finding where the slope hits zero. Add the x-value in the point field before solving.

How Accurate Are the Calculator's Step-by-Step Solutions?

Before relying on any tool for homework or coursework, accuracy is a fair question to ask upfront. This calculator uses a symbolic differentiation engine: it parses your expression into a tree structure and applies differentiation rules directly to that structure, rather than estimating the slope numerically. That means the results are exact, not approximations. One thing worth keeping in mind is that your textbook or instructor may expect the answer written in a different but equivalent form, so it is worth checking that the simplified output matches what is expected. Because the calculator shows its working alongside the final answer, it is also easier to catch a typo in the original input before it throws off the whole result.

What Other Calculus Tools Pair With a Derivative Calculator?

Finding a derivative is rarely the last step in a calculus problem set. The same assignment that has you differentiating a function often also asks for an integral, a limit, an antiderivative, or a value at a specific point, and each of those has its own dedicated calculator built for that job. The sections below cover how each one connects back to what you have just solved here.

How Does an Integral Calculator Relate to a Derivative Calculator?

Integration undoes differentiation, so once you have worked out a derivative, checking the reverse process on a related problem is often the next thing on the list. The integral calculator is built for exactly that: verifying antiderivative work, or solving area-under-the-curve problems where the integral is what you actually need.

What Is the Difference Between a Derivative and an Antiderivative Calculator?

A derivative tells you the rate of change of a function; an antiderivative works backward from that rate of change to reconstruct the original function. Take d/dx(x²) = 2x. Reverse that, and the antiderivative of 2x is x² + C. When that is the direction you need to go, the antiderivative calculator handles it.

How Does a Limit Calculator Help Before You Differentiate?

The formal definition of a derivative is actually built from a limit, so if you are learning the concept itself rather than just applying the shortcut rules, working through a limit first can make the idea click. This matters more for early calculus coursework than for applied problem-solving, but if that is where you are, the limit calculator walks through that exact step.

When Do You Need an Implicit Differentiation Calculator Instead?

Everything covered above assumes y is written explicitly in terms of x. But equations like x² + y² = 25 do not isolate y easily, and standard differentiation does not apply cleanly to them. That case calls for a different technique and a different input format, which is exactly what the implicit differentiation calculator is set up to handle.

How Does a Partial Derivative Calculator Differ From This One?

This calculator is built for single-variable functions. Once you are working with a function of two or more variables, something like f(x,y), you need partial derivatives instead, which hold every other variable constant while differentiating with respect to just one. That is mostly territory for multivariable calculus, economics, and physics coursework, and the partial derivative calculator is the tool built for it.

How Does a Tangent Line Calculator Use Derivative Results?

This connects directly back to the point-evaluation feature covered earlier on this page. Once you have the derivative at a point, that number is the slope, and the tangent line calculator takes that slope plus the coordinates of the point to produce the full equation of the line, not just the slope value on its own.